$, with $=\Omega^{-1} \int^{x_0+\Omega}_{x_0} q(x) dx$, that is, \begin{equation}\label{4.14} \sigma(H)=\left(\bigcup^{\tilde{g}}_{n=1} \tilde{\sigma}_n \right) \cup \tilde{\sigma}_{\infty}, \end{equation} where each $\tilde{\sigma}_n$ and $\tilde{\sigma}_{\infty}$ is a union of some of the spectral arcs $\sigma_n$ in (\ref{2.22}). \end{theorem} Note that the set $B$ of values of $E$ where $p(E)-2d_i(E)>0$ contains $B_1$, the set of all those points where only less than two linearly independent Floquet solutions exist. For $B\backslash B_1$ to be nonempty, it is necessary that $p(z)\geq3$ for some (anti)periodic eigenvalue $z$. While it seems difficult to construct an explicit example where $B\backslash B_1\neq \emptyset$, the very existence of this phenomenon has first been noted in \cite{32a}. References \cite{24}, \cite{25}, \cite{33}, \cite{50}, \cite{51} treat potentials with $p(E)\leq 2$ and references \cite{9}, \cite{10} require that algebraic and geometric multiplicities of all (anti)periodic eigenvalues coincide and hence also that $p(E) \leq 2$. Generically one has $p(E)-2d_i(E)=1$ if this is positive at all and $B=B_1$ (cf. \cite{60}). \begin{remark}[Singularity structure of the Green's function] As Theorem \ref{t2.2b} shows, it is precisely the multiplicity $p-2d_i$ of the branch and singular points in the Burchnall-Chaundy polynomial (\ref{4.13}) which determines the singularity structure of the diagonal Green's function $G(E,x,x)$ of $H$. Moreover, since (see, e.g., \cite{47}) \begin{equation}\label{2.79} G(E,x,x')=[G(E,x,x)G(E,x',x')]^{1/2} \exp [-\dfrac{1}{2} \int^{\max (x,x')}_{\min(x,x')} G(E,s,s)^{-1}ds], \end{equation} this observation extends to the off-diagonal Green's function $G(E,x,x')$ of $H$ as well. \end{remark} \begin{remark}[Inverse square singularities] The case of the Lam\'e-Ince potential, where $q$ has singularities of the form $-g(g+1)/x^2$, indicates the necessity to consider also potentials with such singularities. This is possible by modifying the usual approach via Volterra integral equations which are used to obtain the asymptotic properties (\ref{2.32})--(\ref{2.33}) of the corresponding eigenvalue distributions. One obtains essentially the same results as in the present review, the only difference being that the conditional stability set cannot be interpreted as the spectrum of an operator in $L^2(\bb R)$. This approach has been worked out in detail in \cite{w2} and \cite{w3}. \end{remark} \begin{remark}[Finite-band potentials] For real-valued potentials Novikov \cite{48} and Dubrovin \cite{15} showed that $q$ is an algebro-geometric potential if and only if the spectrum of the operator $H$ consists of only finitely many bands. This is no longer true for complex-valued potentials. In fact, for $q=\e^{ix}$ one infers $\sigma(H)=(-\infty,0]$ but every Dirichlet eigenvalue is movable (see \cite{w3}). \end{remark} \section{A Characterization of Elliptic Solutions of the KdV Hierarchy} \label{picard} \setcounter{equation}{0} In this section we discuss the principal result in \cite{32a}, an explicit characterization of all elliptic algebro-geometric solutions of the KdV hierarchy. One of the two key ingredients in our main Theorem \ref{t3.7} (the other being Theorem \ref{t2.2a}) is a systematic use of a powerful theorem of Picard (see Theorem \ref{t3.1} below) concerning the existence of solutions which are elliptic functions of the second kind of ordinary differential equations with elliptic coefficients. We start with Picard's theorem. \begin{theorem}\label{t3.1} (\cite{63}, \cite{64}, \cite{65}, see, e.g., \cite{3}, p. 182--187, \cite{37}, p. 375--376) Let $q_m$, $1\leq m \leq n$ be elliptic functions with a common period lattice spanned by the fundamental periods $2\omega_1$ and $2\omega_3$. Consider the differential equation \begin{equation}\label{3.1} \sum^n_{m=0} q_m(z)\psi^{(m)}(z)=0, \; q_n(z)=1, \; z\in\bb C \end{equation} and assume that (\ref{3.1}) has a meromorphic fundamental system of solutions. Then there exists at least one solution $\psi_0$ which is elliptic of the second kind, that is, $\psi_0$ is meromorphic and \begin{equation}\label{3.2} \psi_0(z+2\omega_j)=\rho_j\psi_0(z), \; j=1,3 \end{equation} for some constants $\rho_1, \rho_3\in\bb C$. If in addition, the characteristic equation corresponding to the substitution $z\to z+2\omega_1$ or $z\to z+2\omega_3$ (see \cite{37}, p. 358, 376) has distinct roots then there exists a fundamental system of solutions of (\ref{3.1}) which are elliptic functions of the second kind. \end{theorem} The characteristic equation associated with the substitution $z\to z+2\omega_j$ alluded to in Theorem \ref{t3.1} is given by \begin{equation}\label{3.3} \det [A-\rho I]=0, \end{equation} where \begin{equation}\label{3.4} \phi_{\ell}(z+2\omega_j) = \sum^n_{m=1} a_{\ell,m}\phi_m(z), \; A=(a_{\ell,m})_{1\le \ell, m \le n} \end{equation} and $\phi_1,....,\phi_n$ is any fundamental system of solutions of (\ref{3.1}). What we call Picard's theorem following the usual convention in \cite{3}, p. 182--185, \cite{13}, p. 338--343, \cite{34}, p. 536--539, \cite{41}, p. 181--189, appears, however, to have a longer history. In fact, Picard's investigations \cite{53}, \cite{54}, \cite{55} were inspired by earlier work of Hermite in the special case of Lam\'e's equation \cite{35}, p. 118--122, 266--418, 475--478 (see also \cite{6a}, Sect. 3.6.4 and \cite{67}, p. 570--576). Further contributions were made by Mittag-Leffler \cite{46}, and Floquet \cite{21}, \cite{22}, \cite{23}. Detailed accounts on Picard's differential equation can be found in \cite{34}, p. 532--574, \cite{41}, p. 198--288. In this context it seems appropriate to recall the well-known fact (see, e.g., \cite{3}, p. 185--186) that $\psi_0$ is elliptic of the second kind if and only if it is of the form \begin{equation}\label{3.5} \psi_0(z)=Ce^{\lambda z} \prod^m_{j=1}[\sigma(z-a_j)/\sigma(z-b_j)] \end{equation} for suitable $m\in\bb N$ and $C,\lambda,a_j,b_j \in\bb C$, $1\le j\le m$. Here $\sigma(z)$ is the Weierstrass sigma function associated with the period lattice $\Lambda$ spanned by $2\omega_1, 2\omega_3$ (see \cite{1}, Ch. 18). Picard's Theorem \ref{t3.1}, restricted to the second-order case \begin{equation}\label{3.6} \psi''(z) + q(z)\psi(z)=E\psi(z), \end{equation} motivates the following definition. \begin{definition} Let $q$ be an elliptic function. Then $q$ is called a {\bf Picard potential} if and only if the differential equation (\ref{3.6}) has a meromorphic fundamental system of solutions (with respect to $z$) for each value of the spectral parameter $E\in\bb C$. \end{definition} For completeness we recall the following result. \begin{theorem}(\cite{31}) \label{p3.3} (i) Any non-constant Picard potential $q$ has a representation of the form \begin{equation} \label{3.7} q(z)=C-\sum^m_{j=1} s_j(s_j+1) \wp(z-b_j) \end{equation} for suitable $m, s_j\in\bb N$ and $C, b_j \in \bb C$, $1\le j\le m$, where the $b_j$ are pairwise distinct $\operatorname{mod} (\Lambda)$ and $\wp(z)$ denotes the Weierstrass $\wp$-function associated with the period lattice $\Lambda$ (\cite{1}, Ch. 18).\\ (ii) Let $q(z)$ be given as in (\ref{3.7}). If $\psi^{''}+q\psi=E\psi$ has a meromorphic fundamental system of solutions for a number of distinct values of $E$ which exceeds $\max\{s_1,\ldots,s_m\},$ then $q$ is a Picard potential. \end{theorem} We emphasize that while any Picard potential is necessarily of the form (\ref{3.7}), a potential $q$ of the type (\ref{3.7}) is a Picard potential only if the constants $b_j$ satisfy a series of additional intricate constraints, see, for instance, Section 3.2 in \cite{31}. The following result indicates the connection between Picard potentials and elliptic algebro-geometric potentials. \begin{theorem}\label{t3.4} (Its and Matveev \cite{39}, Krichever \cite{42}, \cite{43}, Segal and Wilson \cite{57}) Every elliptic algebro-geometric potential $q$ is a Picard potential. \end{theorem} \begin{proof}[Sketch of Proof] For nonsingular curves $K_g:F^2=\prod^{2g}_{j=0}(E-\hat{E}_j)$ associated with $q$ (see (\ref{2.16})), where $\hat{E}_{\ell}\ne \hat{E}_{\ell'}$ for $\ell\ne \ell'$, Theorem \ref{t3.4} is obvious from the standard representation of the Baker-Akhiezer function in terms of the Riemann theta function of $K_g$ (\cite{16}, \cite{39}, \cite{42}, \cite{43}). For singular curves $K_g$ the result follows from the $\tau$-function representation of the Floquet solutions $\psi_{\pm}(E,x)$ associated with $q$ \begin{equation}\label{3.8} \psi_{\pm}(E,x)=e^{\pm k(E)x} \tau_{\pm}(E,x) /\tau(x), \end{equation} where \begin{equation}\label{3.9} q(x)=C+2\{\ln[\tau(x)]\}'' \end{equation} and from the fact that $\tau(x)$ and $\tau_{\pm}(E,x)$ are entire with respect to $x$ (cf. \cite{57}). \end{proof} Naturally, one is tempted to conjecture that the converse of Theorem \ref{t3.4} is true as well. The rest of this section will explain our proof of this conjecture in \cite{32a}. We start with a bit of notation. Let $q(z)$ be an elliptic function with fundamental periods $2\omega_1, 2\omega_3$ and assume, without loss of generality, that $\re(\omega_1)>0$, $\re(\omega_3) \ge 0$, $\im(\omega_3/\omega_1)>0$. The fundamental period parallelogram then consists of the points $z=2\omega_1 s +2\omega_3 t$, where $0\leq s,t<1$. We introduce \begin{equation}\label{3.11} e^{i\phi} = \dfrac{\omega_3}{\omega_1} \left| \dfrac{\omega_1}{\omega_3} \right|, \; \phi \in (0,\pi), \end{equation} and \begin{equation}\label{3.12} t_j=\omega_j/|\omega_j|, \; j=1,3 \end{equation} and define \begin{equation}\label{3.13} q_j(x):=t^2_j q (t_jx+z_0), \; j=1,3 \end{equation} for a $z_0\in\bb C$ which we choose in such a way that no pole of $q_j, j=1,3$ lies on the real axis. (This is equivalent to the requirement that no pole of $q$ lies on the line through the points $z_0$ and $z_0+2\omega_1$ or on the line through $z_0$ and $z_0+2\omega_3.$ Since $q$ has only finitely many poles in the fundamental period parallelogram this can always be achieved.) For such a choice of $z_0$ we infer that $q_j(x)$ are real-analytic and periodic of period $\Omega_j=2|\omega_j|, \; j=1,3.$ Comparing the differential equations \begin{equation}\label{3.14} \psi''(z)+q(z)\psi(z)=E\psi(z) \end{equation} and \begin{equation}\label{3.15} w''(x) + q_j(x)w(x) = \lambda w(x), \; j=1,3, \end{equation} connected by the variable transformation \begin{equation}\label{3.16} z=t_jx+z_0, \; \psi(z) = w(x), \end{equation} one concludes that $w$ is a solution of (\ref{3.15}) if and only if $\psi$ is a solution of (\ref{3.14}) with \begin{equation}\label{3.17} \lambda=t_j^2E, \; j=1,3. \end{equation} Next, consider $\tilde{q}\in C^0(\bb R)$ of period $\tilde{\Omega} >0$ and let $\tilde{c}(\lambda,x), \tilde{s}(\lambda,x)$ be the corresponding fundamental system of solutions of $\tilde{w}^{''}+ \tilde{q}\tilde{w}= \lambda\tilde{w}$ defined by \begin{equation}\label{3.18} \tilde{c}(\lambda,0) = \tilde{s}'(\lambda,0)=1, \quad \tilde{c}'(\lambda,0) = \tilde{s}(\lambda,0)=0. \end{equation} The corresponding Floquet discriminant is now given by \begin{equation}\label{3.19} \tilde{\Delta}(\lambda)=[\tilde{c}(\lambda,\tilde{\Omega}) + \tilde{s} (\lambda,\Omega)]/2 \end{equation} and Rouch\'e's theorem then yields \begin{equation}\label{3.20} \tilde{\Delta}(\lambda)= \cos[i\tilde{\Omega}\lambda^{1/2}(1+O(\lambda^{-1}))] \end{equation} as $|\lambda|$ tends to infinity. \begin{lemma}\label{p3.5} Let $\tilde{\lambda}_n$ be a periodic or antiperiodic eigenvalue of $\tilde{q}$. Then there exists an $m\in\bb Z$ such that \begin{equation}\label{3.21} \left|\tilde{\lambda}_n + m^2\pi^2 \tilde{\Omega}^{-2}\right|\le \tilde{C} \end{equation} for some $\tilde{C}>0$ independent of $n\in\bb N_0$. In particular, all periodic and antiperiodic eigenvalues $\tilde{\lambda}_n$, $n\in\bb N_0$ of $\tilde{q}$ are contained in a half-strip $\tilde{S}$ given by \begin{equation}\label{3.22} \tilde{S}=\{\lambda\in\bb C | \left|\im(\lambda)\right|\le \tilde{C}, \; \re(\lambda)\le \tilde{M}\} \end{equation} for some $\tilde{M}\in\bb R.$ \end{lemma} In order to apply Lemma \ref{p3.5} to $q_1$ and $q_3$ we note that according to (\ref{3.20}), \begin{equation}\label{3.32} \Delta_j(\lambda)=\cos [i\Omega_j\lambda^{1/2} (1+O(\lambda^{-1}))], \; j=1,3 \end{equation} as $|\lambda|$ tends to infinity, where, in obvious notation, $\Delta_j(\lambda)$ denotes the discriminant of $q_j(x)$, $j=1,3$. Next, denote by $\lambda_{j,n}$ an $\Omega_j$-(anti)periodic eigenvalue of $w^{''}+q_jw=\lambda w.$ Then $E_{j,n} = t_j^{-2}\lambda_{j,n}$ is a $2\omega_j$-(anti)periodic eigenvalue of $\psi^{''}+ q \psi = E\psi$ and vice versa. Hence Lemma \ref{p3.5} immediately yields the following result. \begin{lemma}\label{p3.6} Let $j=1$ or $3$. Then all $2\omega_j$-(anti)periodic eigenvalues $E_{j,n}$, $n\in\bb N_0$ associated with $q$ lie in the half-strip $S_j$ given by \begin{equation}\label{3.33} S_j=\{E\in\bb C: |\im(t^2_jE)|\le C_j, \; \re (t^2_jE) \le M_j\} \end{equation} for suitable constants $C_j>0, M_j\in\bb R.$ The angle between the axes of the strips $S_1$ and $S_3$ equals $2\phi \in (0,2\pi).$ \end{lemma} Lemmas \ref{p3.5} and \ref{p3.6} apply to any elliptic potential whether or not they are algebro-geometric. In our final step we shall now invoke Picard's Theorem \ref{t3.1} to obtain our characterization of elliptic algebro-geometric potentials. \begin{theorem}\label{t3.7} $q$ is an elliptic algebro-geometric potential if and only if $q$ is a Picard potential (i.e., if and only if for each $E\in\bb C$ every solution of $\psi''(z)+ q(z)\psi(z)=E\psi(z)$ is meromorphic with respect to $z$). \end{theorem} \begin{proof} By Theorem \ref{t3.4} it remains to prove that a Picard potential is algebro-geometric. Hence we assume in the following that $q$ is a Picard potential. Since all $2\omega_j$-(anti)periodic eigenvalues $E_{j,n}$ of $q$ yield zeros $\lambda_{j,n} = t^2_jE_{j,n}$ of the entire functions $\Delta_j(\lambda)^2-1$, the $E_{j,n}$ have no finite limit point. Next we choose $R>0$ large enough such that the exterior of the closed disk $\overline{D(0,R)}$ centered at the origin of radius $R>0$ contains no intersection of $S_1$ and $S_3$ (defined in (\ref{3.33})), that is, \begin{equation}\label{3.34} (\bb C\backslash \overline{D(0,R)}) \cap (S_1\cap S_3) = \emptyset. \end{equation} Let $\rho_{j,\pm}(\lambda)$ be the Floquet multipliers of $q_j(x),$ that is, the solutions of \begin{equation}\label{3.35} \rho^2_j-2\Delta_j \rho_j+1=0, \; j=1,3. \end{equation} Then (\ref{3.34}) implies that for $E\in\bb C\backslash \overline{D(0,R)}$, at most one of the numbers $\rho_1(t_1E)$ and $\rho_3(t_3E)$ can be in $\{-1,1\}$. In particular, at least one of the characteristic equations corresponding to the substitution $z\to z+2\omega_1$ or $z\to 2\omega_3$ (cf. (\ref{3.3}) and (\ref{3.4})) has two distinct roots. Since by hypothesis $q$ is a Picard potential, Picard's Theorem \ref{t3.1} applies and guarantees for all $E\in \bb C\backslash \overline{D(0,R)}$ the existence of two linearly independent solutions $\psi_1(E,z)$ and $\psi_2(E,z)$ of $\psi^{''} + q\psi= E\psi$ which are elliptic of the second kind. Then $w_{j,k}(x)= \psi_k(t_j x+z_0)$, $k=1,2$ are linearly independent Floquet solutions associated with $q_j$. Therefore the points $\lambda$ for which $w''+q_jw= \lambda w$ has only one Floquet solution are necessarily contained in $\overline{D(0,R)}$ and hence finite in number. This is true for both $j=1$ and $j=3$. Applying Theorem \ref{t2.2a} then proves that both $q_1$ and $q_3$ are algebro-geometric potentials. By (\ref{2.8}) (in slight abuse of notation) \begin{equation} \sum^{g}_{k=0}c_{g-k}\frac{df_{k+1}(q_1(x))}{ dx}=0, \end{equation} where $g\in\bb N_0$, $f_{k+1}$, $k=0,...,g$, are differential polynomials in $q_1$ homogeneous of degree $2k+2$ (cf. (\ref{2.9})), and $c_{k}$, $k=0,...,g$ are complex constants. Since \begin{equation}\label{3.37} q_1^{(\ell)}(x)=t_1^{\ell+2} q^{(\ell)}(z), \end{equation} (where $z=t_1x+z_0$) we obtain \begin{equation} \sum^{g}_{k=0}c_{g-k} t_1^{2k+3} \frac{df_{k+1}(q(z))}{ dz}=0, \end{equation} that is, $q$ is an algebro-geometric potential as well. A similar argument would have worked using the relationship between $q_3$ and $q$. In particular, the order of the operators commuting with $d^2/dz^2+q(z)$, $d^2/dx^2+q_1(x)$, and $d^2/dx^2+q_3(x)$, respectively, is the same in all cases, namely $2g+1$. \end{proof} We add a series of remarks further illustrating the significance of Theorem \ref{t3.7}. \begin{remark} [Complementing Picard's theorem] First we note that Theorem \ref{t3.7} extends and complements Picard's Theorem \ref{t3.1} in the sense that it determines the elliptic functions which satisfy the hypothesis of the theorem precisely as (elliptic) algebro-geometric solutions of the stationary KdV hierarchy. \end{remark} \begin{remark}[Characterizing elliptic algebro-geometric potentials] While an explicit proof of the algebro-geometric property of $q$ is, in general, highly nontrivial (see, e.g., the references cited in connection with special cases such as the Lam\'e-Ince and Treibich-Verdier potentials in Remark \ref{r3.12} below), the fact of whether or not $\psi^{''}(z)+q(z)\psi(z)=E\psi(z)$ has a fundamental system of solutions meromorphic in $z$ for a finite (but sufficiently large) number of values of the spectral parameter $E\in\bb C$ can be decided by means of an elementary Frobenius-type analysis (see, e.g., \cite{29} and \cite{30}). Theorem \ref{t3.7} appears to be the only effective tool to identify general elliptic algebro-geometric solutions of the KdV hierarchy. \end{remark} \begin{remark} [Reduction of Abelian integrals] Theorem \ref{t3.7} is also relevant in the context of the Weierstrass theory of reduction of Abelian to elliptic integrals, a subject that attracted considerable interest, see, for instance, \cite{5}, \cite{6}, \cite{6a}, Ch. 7, \cite{7}, \cite{8}, \cite{13d}, \cite{18}, \cite{19}, \cite{20}, \cite{38}, \cite{40}, \cite{44}, \cite{44c}, \cite{58d}, \cite{58}, \cite{59}. In particular, the theta functions corresponding to the hyperelliptic curves derived from the Burchnall-Chaundy polynomials (\ref{2.15}), associated with Picard potentials, reduce to one-dimensional theta functions. \end{remark} \begin{remark} [Computing genus and branch points] \label{r3.12} Even though Theorem \ref{t3.7} characterizes all elliptic algebro-geometric potentials as Picard potentials, it does not yield an effective way to compute the underlying hyperelliptic curve $K_g$; in particular, its proof provides no means to compute the branch and singular points nor the (arithmetic) genus $g$ of $K_g$. To the best of our knowledge $K_g$ has been computed only for Lam\'e-Ince potentials and certain Treibich-Verdier potentials (see, e.g., \cite{4a}, \cite{8}, \cite{40}, \cite{49}, \cite{58}, \cite{59}, \cite{64}, \cite{66}, \cite{67}). Even the far simpler task of computing $g$ previously had only been achieved in the case of Lam\'e-Ince potentials (see \cite{36} and \cite{61} for the real and complex-valued case, respectively). In \cite{29}, \cite{30}, and \cite{31} we have treated these problems for Lam\'e-Ince, Treibich-Verdier, and reflection symmetric elliptic algebro-geometric potentials, respectively. In particular, in \cite{30} we computed $g$ for all Treibich-Verdier potentials and in \cite{31} we reduced the computation of the branch and singular points of $K_g$ for any reflection symmetric elliptic algebro-geometric potential to the solution of constraint linear algebraic eigenvalue problems. We refrain from reproducing a detailed discussion of this matter here, instead we just recall an example taken from \cite{30} which indicates some of the subtleties involved: Consider the potentials \begin{gather}\label{3.45} q_4(z)=-20\wp(z-\omega_j)-12\wp(z-\omega_k), \displaybreak[0] \\ \hat{q}_4(z)=-20\wp(z-\omega_j)-6\wp(z-\omega_k) -6\wp(z-\omega_{\ell}), \\ q_5(z)=-30\wp(z-\omega_j)-2\wp(z-\omega_k), \\ \hat{q}_5(z)= -12\wp(z-\omega_j)-12\wp(a-\omega_k)-6\wp (z-\omega_{\ell})-2\wp(z-\omega_m), \end{gather} where $j,k,\ell,m\in\{1,2,3,4\}$ ($\omega_2=\omega_1+\omega_3$, $\omega_4=0$) are mutually distinct. Then $q_4$ and $\hat{q}_4$ correspond to (arithmetic) genus $g=4$ while $q_5$ and $\hat{q}_5$ correspond to $g=5$. However, we emphasize that all four potentials contain precisely 16 summands of the type -- $2\wp(x-b_n)$ (cf. the discussion following (\ref{1.10})). $q_5$ and $\hat{q}_5$ are isospectral (i.e., correspond to the same curve $K_5$) while $q_4$ and $\hat{q}_4$ are not. \end{remark} \section{Picard's Theorem for First-Order Systems} \label{systems} \setcounter{equation}{0} Having characterized all elliptic algebro-geometric solutions of the KdV hierarchy which are related to the second-order expression $L_2 = d^2/dz^2+q$, it is natural to try to extend Theorem \ref{t3.7} to $n$-th order expressions $L_n$ (connected with the Gel'fand-Dickey hierarchy). Actually, a more general extension to integrable systems related to general first-order $n\times n$ matrix-valued differential expressions seems very desirable in order to include AKNS systems (see, e.g., \cite{27b}) and the matrix hierarchies of integrable equations described in detail, for instance, in \cite{14}, Sects. 9, 13--16, \cite{15a}, \cite{16a}. Picard's Theorem \ref{t3.1} generalizes in a straightforward manner to first-order systems, that is, pairwise distinct Floquet multipliers in one of the fundamental directions and a meromorphic fundamental system of solutions guarantee the existence of a fundamental system of solutions which are elliptic of the second kind (see (\ref{5.5})). Moreover, it is possible to obtain the explicit Floquet-type structure of these solutions (cf. Theorem \ref{t5.3}). Denote by $M(n)$ the set of $n\times n$ matrices with entries in $\bb C$ and consider the linear homogeneous system \begin{equation}\label{5.1} \Psi'(z) = Q(z)\Psi(z), \quad x\in\bb C, \end{equation} where $Q(z)\in M(n)$ and where the entries of $Q(z)$ are elliptic functions with a common period lattice $\Lambda$ spanned by $2\omega_1$ and $2\omega_3$ which satisfy the same conditions as before. Assuming without loss of generality that no pole of $Q(z)$ lies on the line containing the segments $[0,2\omega_j]$, Floquet theory with respect to these directions yields the existence of fundamental matrices $\Phi_j(z)$ of the type \begin{equation}\label{5.3} \Phi_{j}(z) = P_{j}(z)\exp\lp z K_j\rp, \end{equation} where $P_{j}(z)$ is a periodic matrix with period $2\omega_j$ and $K_j$ is a constant matrix. The monodromy matrix is given by $M_j=\exp(2\omega_j K_j)$. We want to establish the existence of a Floquet representation {\it simultaneously} for both directions $2\omega_1$ and $2\omega_3$. More precisely, we intend to find solutions $\underline\phi$ of $\Psi'(z) = Q(z)\Psi(z)$ satisfying \begin{equation}\label{5.5} \underline\phi(z+2\omega_j) = \rho_j\underline\phi, \quad j=1,3, \end{equation} where $\rho_j\in\cmz$. Solutions $\underline\phi(z)$ of (\ref{5.1}) satisfying (\ref{5.5}) are again called elliptic of the second kind. Even though Picard did mention certain extensions of his result to first-order systems (see, e.g., \cite{gray}, p. 248--249), apparently he did not seek a Floquet representation for systems in the elliptic case. The first to study such a representation seems to have been Fedoryuk who proved the following result. \begin{theorem}\label{t5.1}(\cite{fedor}) Let $Q(z)$ be an $n\times n$ matrix whose entries are elliptic functions with fundamental periods $2\omega_1$ and $2\omega_3$ and suppose that (\ref{5.1}) has a single-valued fundamental matrix of solutions. Then (\ref{5.1}) admits a fundamental matrix $\Phi(z)$ of the type \begin{equation}\label{5.6} \Phi(z) = D(z)\exp\lp zS+\zeta(z)T\rp, \quad z\in\bb C, \end{equation} where $S,T\in M(n)$, $D(z)$ is invertible and doubly periodic and \begin{equation}\label{5.8} S = \frac{1}{\pi i} \lb2\omega_3\zeta\lp\omega_1\rp K_3 - 2\omega_1\zeta\lp\omega_3\rp K_1\rb, \quad T = -\frac{2\omega_1 \omega_3}{\pi i}\lp K_3- K_1\rp. \end{equation} Moreover, $K_1$ and $K_3$, and hence $S$ and $T$ commute. \end{theorem} Fedoryuk's representation (\ref{5.6}) has the peculiar feature that it seems to stress an apparent essential singularity structure of solutions at $z=0$. Indeed, since $\zeta(z)$ has a first-order pole at $z=0$, the term $\exp\lp \zeta(z)T\rp$ in (\ref{5.6}) exhibits an essential singularity unless $T$ is nilpotent. Hence, the doubly periodic matrix $D(z)$, in general, will cancel the essential singularity of $\exp\lp \zeta(z)T\rp$ and therefore cannot be meromorphic and hence not elliptic. Thus Fedoryuk's result cannot be considered the natural extension of Picard's Theorem \ref{t3.1}. In the remainder of this section we shall focus on an alternative to Theorem \ref{t5.1} and describe a generalization of Picard's theorem in the context of first-order systems with elliptic coefficients. \begin{theorem}\label{t5.3} (\cite{27a}) Let $Q(z)$ be an elliptic $n\times n$ matrix with fundamental periods $2\omega_1$ and $2\omega_3$ and suppose that (\ref{5.1}) has a meromorphic fundamental matrix $\Psi(z)$ of solutions. Then (\ref{5.1}) admits a fundamental matrix of the type \begin{align}\label{5.16} \Phi(z)=&E(z)\sigma(z)^{-1} \sigma\lp zI_n -\frac{2\omega_1 \omega_3}{\pi i}\lp K_3- K_1\rp\rp\\ &\times \exp\lbr\frac{z}{\pi i} \lb2\omega_3\zeta\lp\omega_1\rp K_3- 2\omega_1\zeta\lp\omega_3\rp K_1\rb\rbr, \quad z\in\bb C, \notag \end{align} where $E(z)$ is an elliptic matrix with periods $2\omega_j$ and $K_1$, $K_3$ (and hence $M_j=\exp\lp2\omega_j K_j\rp$, $ j=1,3$) are commuting matrices. Moreover, linearly independent solutions $\underline\phi_m(z)\in\bb C^n$, $ 1\leq m\leq n$ of (\ref{5.1}), that is, column vectors of (\ref{5.16}), are of the type \begin{equation}\label{5.19} \underline\phi_m(z)=\sum_{k_{1}=0}^{n_{1}}\sum_{k_{2}=0}^{n_{2}} \underline{e}_{m,k_1,k_2}(z)\exp\lp z \mu_{m,k_1,k_2}\rp z^{k_{1}}\zeta(z)^{k_{2}}, \end{equation} where the vectors $\underline{e}_{m,k_1,k_2}(z)$ are elliptic, the numbers $\mu_{m,k_1,k_2}$ denote the (not necessarily distinct) eigenvalues of \begin{equation} \lp 1/\pi i\rp\lb2\omega_3\zeta\lp2\omega_1/2\rp K_3 -2\omega_1\zeta\lp2\omega_3/2\rp K_1\rb, \end{equation} and, most notably, the upper limits of the sums in (\ref{5.19}) satisfy \begin{equation}\label{5.20} n_{1}+n_{2}\leq n-1. \end{equation} In particular, there exists at least one solution $\underline\phi_{m_0}(z)$ of (\ref{5.1}) which is elliptic of the second kind, that is, $\underline\phi_{m_0}(z)$ is meromorphic on $\bb C$ and \begin{equation}\label{5.21} \underline\phi_{m_0}(z+2\omega_j)=\rho_{m_{0},j} \underline\phi_{m_0}(z), \quad j=1,3, \quad z\in\bb C \end{equation} for some $\rho_{m_{0},j} =\exp\lp2\omega_j\mu_{m_{0},0,0}\rp\in\cmz$, $ j=1,3$. In addition, if all eigenvalues of $M_1$ or $M_3$ are distinct, then there exists a fundamental system of solutions $\{\underline\phi_m(z)\}_{ 1\leq m\leq n}$ of (\ref{5.1}) with all $\underline\phi_{m}(z)$ elliptic of the second kind. \end{theorem} For the proof one considers a meromorphic fundamental matrix $\tilde\Psi(z)$ of \ref{5.1} and defines \begin{align}\label{5.23} E(z)=&\tilde\Psi(z)\exp\lp -\frac{z}{\pi i} \lb2\omega_3\zeta\lp\omega_1\rp K_3 -2\omega_1\zeta\lp\omega_3\rp K_1 \rb\rp \\ &\times \sigma\lp z I_n -\frac{2\omega_1 \omega_3}{\pi i}\lp K_3- K_1\rp\rp^{-1}\sigma(z). \notag \end{align} By hypothesis, $E(z)$ is meromorphic and, applying the addition theorem \begin{equation}\label{5.13} \sigma(z+2\omega_j) =-\sigma(z)\exp\left\{2\zeta\lp\omega_j\rp\lb z +\lp\omega_j\rp\rb\right\}, \quad 1\leq j\leq 3 \end{equation} (more precisely, a matrix-valued generalization thereof), one verifies that \begin{equation}\label{5.24} E(z+2\omega_j) = E(z), \quad j=1,3, \end{equation} that is, $E(z)$ is elliptic. The remaining assertions in Theorem \ref{t5.3} follow by transforming $(K_3 -K_1)$ and, say, $K_1$ (separately) into their Jordan normal forms (cf. \cite{27a}). \begin{remark}\label{r5.4} If $M_1$ or $M_3$ has distinct eigenvalues, one can show that equation (\ref{5.1}) has a fundamental system of solutions $\underline\phi_m(z)$ of the form, \begin{align}\label{5.25} \underline\phi_m(z)=&\underline{e}_m(z)\sigma(z)^{-1} \sigma\lp z-\frac{2\omega_1\omega_3}{\pi i}k_{3,1,m}\rp \\ &\times\exp\lp\frac{2\omega_3 z}{\pi i}\zeta\lp\omega_1\rp k_{3,1,m}\rp\exp\lp z k_{1,m}\rp,\quad 1\leq m\leq n, \quad z\in\bb C, \notag \end{align} where $\underline{e}_m(z)$ are elliptic with period lattice $\Lambda$ and $\lbr k_{3,1,m}\rbr_{ 1\leq m\leq n}$ and $\lbr k_{1,m}\rbr_{ 1\leq m\leq n}$ are the eigenvalues of $( K_3- K_1)$ and $K_1$, respectively. \end{remark} \begin{remark}\label{r5.5} In the special scalar case (\ref{3.1}), the bound (\ref{5.20}) was proved in \cite{37}, p. 377--378 for $n=2$ and stated (without proof) for $n\geq 3$. \end{remark} \section{The Higher-Order Scalar Case}\label{nthorder} \setcounter{equation}{0} In our final section we consider a differential expression $L_n$ of the form \begin{equation}\label{6.1} L_n y=y^{(n)}+q_{n-2}y^{(n-2)}+...+q_0y, \end{equation} where initially the coefficients $q_0,...,q_{n-2}$ are continuous complex-valued functions of a real variable periodic with period $\Omega>0$. Again we denote the ($n$-dimensional) vector space of solutions of the differential equation $L_ny=Ey$ by $\el(E)$ and the operator which shifts the argument of a function in $\el(E)$ by a period $\Omega$ by $T(E)$. As before $T(E)$ and $L_n$ commute which implies that $T(E)$ maps $\el(E)$ to itself. Floquet multipliers, that is, eigenvalues of $T(E)$ are given as zeros of the polynomial \begin{equation}\label{6.2} \F(E,\rho)=(-1)^n \rho^n + (-1)^{n-1} a_{1}(E) \rho^{n-1}+ ... -a_{n-1}(E)\rho+1=0, \end{equation} where the functions $a_1,...,a_{n-1}$ are entire. This is obvious after choosing the basis $\phi_1(E,x),...,\phi_n(E,x)$ of $\el(E)$ satisfying the initial conditions $\phi_j^{k-1}(E,x_0)=\delta_{j,k}$. Note that $\F(E,\cdot)$ has $n$ distinct zeros unless the discriminant of $\F(E,\cdot)$, which is an entire function of $E$, is equal to zero. Thus, this happens at most at countably many points. Denote by $m_g(E,\rho)$ and $m_f(E,\rho)$ the geometric and algebraic multiplicity, respectively, of the eigenvalue $\rho$ of $T(E)$. Then the number $m_f(E,\rho)-m_g(E,\rho)\in\{0,1,...,n-1\}$ counts the ``missing'' Floquet solutions of $L_n y=Ey$ with multiplier $\rho$. We will be interested in the case where this number is positive only for finitely many points $E$. For $\theta\in\bb C$, consider the operator $H(\theta)$ associated with the differential expression $L_n$ in $L^2([x_0,x_0+\Omega])$ with domain \begin{equation}\label{6.3} D(H(\theta))=\{g\in H^{2,n}([x_0,x_0 +\Omega]): g^{(k)}(x_0 +\Omega)=e^{i\theta} g^{(k)}(x_0), \, k=0,...,n-1\}. \end{equation} $H(\theta)$ has discrete spectrum. In fact, its eigenvalues, which will be called Floquet eigenvalues, are given as the zeros of $\F(\cdot,\rho)$. Moreover, the algebraic multiplicity $m_a(E,\rho)$ of $E$ as an eigenvalue of $H(\theta)$ is given as the order of $E$ as a zero of $\F(\cdot,\rho)$ (see, e.g., \cite{32}). With any differential expression $L_n$ given by (\ref{6.1}) with continuous complex-valued periodic coefficients $q_0,...,q_{n-2}$ of a real variable we associate the corresponding closed operator $H: H^{2,n}\to L^2(\bb R)$, $Hy=L_n y$. Rofe-Beketov's result \cite{56}, referred to in Section \ref{floquet}, originally was proved for an $n$-th order operator. Hence the spectrum $\sigma(H)$ of $H$ equals the conditional stability set ${\mathcal S}(L_n)$ of $L_n$, that is, the set of all complex numbers $E$ for which the differential equation $L_n y=Ey$ has a nontrivial bounded solution. For $E$ to be in ${\mathcal S}(L_n)$ it is necessary and sufficient that $L_n y=Ey$ has a Floquet multiplier of modulus one. Hence \begin{equation}\label{6.4} {\mathcal S}(L_n)=\{E\in\bb C: {\F}(E,e^{it})=0 \text{ for some $t\in\bb R$}\}, \end{equation} where $\F$ is given by (\ref{6.2}). Since $\F$ is entire in both its variables, it follows that $\sigma(H)={\mathcal S}(L_n)$ consists of (generally) infinitely many regular analytic arcs (i.e., spectral bands). They end at a point where the arc fails to be regular analytic or extends to infinity. Finite endpoints of the spectral bands are called band edges. \begin{definition}\label{d6.1} (i) $H$ is called a {\bf finite-band operator} if and only if $\sigma(H)$ consists of a finite number of regular analytic arcs.\\ (ii) $L_n$ is called a {\bf Picard differential expression} if and only if all its coefficients are elliptic functions associated with a common period lattice and if $L_n y=Ey$ has a meromorphic fundamental system (with respect to the independent variable) for any value of the spectral parameter $E \in \bb C$. \end{definition} Next, let $\Phi(E,x)$ be the fundamental matrix of $L_n y=Ey$ satisfying the initial condition $\Phi(E,x_0)=I_n$ where $I_n$ is the $n\times n$ identity matrix. The Floquet multipliers of the differential equation $L_n y=Ey$ are then the eigenvalues of the monodromy matrix $\Phi(E,x_0+\Omega)$. Our aim is to determine multiplicities of Floquet eigenvalues and multipliers for large values of the spectral parameter $E$. For large values of $E$ the equation $L_n y=Ey$ can be treated as a perturbation of $y^{(n)} = Ey$. In this case there exist $n$ linearly independent Floquet solutions $\exp(\lambda \sigma_k x)$ with associated Floquet multipliers $\exp(\lambda \sigma_k)$, where $\lambda$ is such that $\lambda^n=-E$ and the $\sigma_k$ are the different $n$-th roots of $-1$. The characteristic polynomial of the associated monodromy matrix is therefore given by \begin{equation}\label{6.5} \F_0(E,\rho)=(-1)^n \rho^n +(-1)^{n-1} a_{F,1}(E)\rho^{n-1}+ ... -a_{F,n-1}(E) \rho + 1, \end{equation} where the $a_{F,j}$ are the elementary symmetric polynomials in the variables $\exp(\lambda \sigma_1)$, ..., $\exp(\lambda \sigma_n)$. Perturbation theory now yields that the coefficients $a_j$ in \eqref{6.2} are related to the coefficients $a_{F,j}$ in \eqref{6.5} by \begin{equation} \label{6.11} a_j(E)=a_{F,j}(E)+b_j(E), \quad j=1,...n-1, \end{equation} where, for some suitable positive constant $M$, \begin{equation}\label{6.12} |b_j| \leq \frac{M}{|\lambda|} |\exp(\lambda\sigma_{n+1-j})... \exp(\lambda\sigma_n)|, \quad j=1,...,n-1 \end{equation} having ordered the roots in such a way that $|\exp(\lambda\sigma_j)| \leq |\exp(\lambda\sigma_{j+1})|$ for $j=1,...,n-1$. This allows one to show that asymptotically, in a certain small disk about $\exp(\lambda\sigma_k)$, there are either one or two Floquet multpliers of $L_ny=Ey$ depending on whether or not the inequality \begin{equation}\label{6.14} \left|\exp(\lambda\sigma_k) - \exp(\lambda\sigma_j)\right| \geq \gamma \max\{|\exp(\lambda\sigma_k)|, |\exp(\lambda\sigma_j)|\} \text{ for all $j\neq k$} \end{equation} holds. From this one may prove the following theorem concerning algebraic multplicities of Floquet multipliers. \begin{theorem}\label{t6.4} (\cite{66a}) Let $L_n$ be defined as in (\ref{6.1}). For every $\varepsilon>0$ there exists a disk $B(\varepsilon)\subset \bb C$ with the following two properties.\\ 1. All values of $E$, where at least two Floquet multipliers of the differential equation $L_n y=Ey$ coincide, lie in $B(\varepsilon)$ or in the cone $\{E:|\im(E)|/|\re(E)|\leq \varepsilon\}$.\\ 2. Every degenerate Floquet multiplier outside $B(\varepsilon)$ has multiplicity two. \end{theorem} This result has been obtained earlier by McKean \cite{44d} for $n=3$ and by da Silva Menezes \cite{13g} for general $n$. Moreover the above observations may be used to obtain information concerning algebraic multiplicities of Floquet eigenvalues. \begin{theorem}(\cite{66a})\label{t6.5} Let $\theta_0\in\bb C$. Then there exists an $R>0$ such that every eigenvalue $E$ of the Floquet operator $H(\theta_0)$, which satisfies $|E|>R$, has at most algebraic multiplicity two. \end{theorem} Now suppose that $\mathcal F(E,\rho)=0$ and that $E\in\mathcal S(L_n)$ is suitably large. Since $1\leq m_f(E,\rho), m_a(E,\rho)\leq2$ we have to distinguish four cases and Weierstrass's preparation theorem provides us with the following information:\\ 1. If $m_f(E,\rho)=m_a(E,\rho)=1$ then one spectral band passes through $E$.\\ 2. If $m_f(E,\rho)=2$ and $m_a(E,\rho)=1$ then two (possibly coinciding) spectral bands end in $E$. \\ 3. If $m_f(E,\rho)=1$ and $m_a(E,\rho)=2$ then two spectral bands intersect in $E$ forming a right angle.\\ 4. If $m_f(E,\rho)=m_a(E,\rho)=2$ then two (possibly coinciding) spectral bands pass through $E$. This shows that a necessary condition for a suitably large $E$ to be a band edge is that $m_a(E,\rho)=1$ and $m_f(E,\rho)=2$ for some $\rho$. In particular, such an $E$ is necessarily a point where strictly less than $n$ linearly independent Floquet solutions exist. In addition, when $n$ is odd then $\sigma(H)$ is ultimately in a cone with the imaginary axis as symmetry axis while the possible band edges (where $m_f(E,\rho)=2$) are in a cone whose axis is the real axis. Therefore we have the following result. \begin{theorem} \label{t6.7a} (\cite{66a}) The operator $H$ associated with the differential expression $L_n$ introduced after (\ref{6.3}) is a finite-band operator whenever $n$, the order of $L_n$, is odd. \end{theorem} Finally we turn to the case where $L_n$ is a Picard differential expression. The principal result of this section, Theorem \ref{t6.8} below, then shows that algebraic and geometric multiplicities of Floquet multipliers of $L_n y=Ey$ can be different only when $E$ is one of finitely many points. \begin{theorem}(\cite{66a})\label{t6.8} Suppose the differential expression $L_n$ is Picard. Then there exist $n$ linearly independent solutions of $L_ny=Ey$ which are elliptic of the second kind for all but finitely many values of the spectral parameter $E$. \end{theorem} \begin{proof}[Sketch of proof] The proof is modeled closely after the one of Theorem \ref{3.7}. Again, inside a compact set there can be only a finite number of values of $E$ where Floquet multipliers associated with a fundamental period of the coefficients of $L_n$ are degenerate. On the other hand when $|E|$ becomes large we only have to prove that for one of the fundamental periods of the coefficients of $L_n$ all Floquet multipliers of $L_n y=Ey$ are distinct according to Picard's Theorem \ref{t3.1}. Assume that the fundamental periods $2\omega_1$ and $2\omega_3$ are such that the angle $\phi$ between them is less than $\pi/n$ and assume that $z_0$ is such that no singularity of $q_0,..., q_{n-2}$ lies on the line through $z_0$ and $z_0+2\omega_1$ or on the line through $z_0$ and $z_0+2\omega_3$. Substituting $w(x)=y(2\omega_1 x+z_0)$ and defining $p_k(x) =(2\omega_1)^{n-k} q_k(2\omega_1 x+z_0)$ transforms $L_ny=Ey$ into \begin{equation}\label{6.15} w^{(n)} + p_{n-2}(x) w^{(n-2)} + ... + p_0(x) y = (2\omega_1)^n E w. \end{equation} Therefore, Theorem \ref{t6.4} implies that all Floquet multipliers associated with the periods $2\omega_1$ ($2\omega_3$) are pairwise distinct provided the spectral parameter $(2\omega_j)^n E$ lies outside the set $S_j$, $j=1,3$, where \begin{equation}\label{6.16} S_j =\left\{z: \left|\frac{\im(z)}{\re(z)}\right| \leq \frac{\phi}{3} \right\} \cup \left\{z: |z| \leq R_j\right\}, \end{equation} with $R_j$, $j=1,3$ being suitable positive constants. The two sets $S_1$ and $S_3$ do not intersect outside a sufficiently large disk $D$. Hence, for each value of $E$ outside $D$, Picard's theorem guarantees the existence of $n$ linearly independent solutions of $L_n y=Ey$ which are elliptic functions of the second kind. \end{proof} In particular, when $|E|$ is large and $L_n$ is Picard, we infer that $m_f(E,\rho) =m_g(E,\rho) \leq m_a(E,\rho)$. Moreover, we have shown earlier that necessarily $m_a(E,\rho)=1$ and $m_f(E,\rho)=2$ for band edges $E$ with $|E|$ sufficiently large. Thus there are no band edges with sufficiently large absolute values for Picard expressions. One may also show that at most two bands extend to infinity. Hence we have the following final theorem which, in view of Theorem \ref{t6.7a}, has significance only when $n$ is even. \begin{theorem}(\cite{66a})\label{t6.9} Let $L_n$ be a Picard differential expression and $H$ the associated operator. Then, if $\sigma(H)$ does not contain closed regular analytic arcs, $\sigma(H)$ consists of finitely many analytic arcs which are regular in their interior. \end{theorem} {\bf Acknowledgments.} F.G. would like to thank all organizers of the AMS meeting at LSU for creating a most stimulating atmosphere and especially for the extraordinary hospitality extended to him. \begin{thebibliography}{999} \bibitem{1} M. Abramowitz and I. A. Stegun, \textit{Handbook of Mathematical Functions}, Dover, New York, 1972. \bibitem{2} H. Airault, H. P. McKean, and J. Moser, \textit{Rational and elliptic solutions of the Korteweg-deVries equation and a related many-body problem}, Commun. Pure Appl. Math. {\bf 30} (1977), 95--148. \bibitem{2b} N. I. Akhiezer, \textit{On the spectral theory of Lam\'{e}'s equation}, Istor.-Mat. Issled {\bf 23} (1978) 77--86, 357. (Russian). \bibitem{3} \bysame, \textit{Elements of the Theory of Elliptic Functions }, Amer. Math. Soc., Providence, RI, 1990. \bibitem{4} S. I. Al'ber, \textit{Investigation of equations of Korteweg-de Vries type by the method of recurrence relations}, J. London Math. Soc. {\bf 19} (1979), 467--480. (Russian) \bibitem{4a} F. M. Arscott, \textit{Periodic Differential Equations}, MacMillan, New York, 1964. \bibitem{5} M. V. Babich, A. I. Bobenko, and V. B. Matveev, \textit{Reductions of Riemann theta-functions of genus $g$ to theta-functions of lower genus, and symmetries of algebraic curves}, Sov. Math. Dokl. {\bf 28} (1983), 304--308. \bibitem{6} \bysame, \textit{Solutions of nonlinear equations integrable in Jacobi theta functions by the method of the inverse problem, and symmetries of algebraic curves }, Math. USSR Izv. {\bf 26} (1986), 479--496. \bibitem{6a} E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A.R. Its, and V. B. Matveev, \textit{Algebro-Geometric Approach to Nonlinear Integrable Equations}, Springer, Berlin, 1994. \bibitem{7} E. D. Belokolos, A. I. Bobenko, V. B. Matveev, and V. Z. Enol'skii, \textit{Algebraic-geometric principles of superposition of finite-zone solutions of integrable non-linear equations}, Russian Math. Surv. {\bf 41:2} (1986), 1--49. \bibitem{8} E. D. Belokolos and V. Z. Enol'skii, \textit{Verdier elliptic solitons and the Weierstrass theory of reduction}, Funct. Anal. Appl. {\bf 23} (1989), 46--47. \bibitem{8a} \bysame, \textit{ Reduction of theta functions and elliptic finite-gap potentials}, Acta Appl. Math. {\bf36} (1994), 87--117. \bibitem{8b} D. Bennequin, \textit{Hommage \`a Jean--Louis Verdier: au jardin des syst\`emes int\'egrables }, Integrable Systems: The Verdier Memorial Conference (ed. by O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach), Birkh\"auser, Boston, 1993, 1--36. \bibitem{9} B. Birnir, \textit{Complex Hill's equation and the complex periodic Korteweg-de Vries equations }, Commun. Pure Appl. Math. {\bf 39} (1986), 1--49. \bibitem{10} \bysame, \textit{Singularities of the complex Korteweg-de Vries flows}, Commun. Pure Appl. Math. {\bf 39} (1986), 283--305. \bibitem{11} J. L. Burchnall and T. W. Chaundy, \textit{Commutative ordinary differential operators}, Proc. London Math. Soc. Ser. 2, {\bf 21} (1923), 420--440. \bibitem{12}\bysame, \textit{Commutative ordinary differential operators}, Proc. Roy. Soc. London {\bf A 118} (1928), 557--583. \bibitem{13} H. Burkhardt, \textit{Elliptische Funktionen}, Verlag von Veit, Leipzig, 2nd ed., 1906. \bibitem{13a} F. Calogero, \textit{Exactly solvable one-dimensional many-body problems }, Lett. Nuovo Cim. {\bf 13} (1975), 411--416. \bibitem{13c} D. V. Choodnovsky and G. V. Choodnovsky, \textit{Pole expansions of nonlinear partial differential equations}, Nuovo Cim. {\bf 40B} (1977), 339--353. \bibitem{13d} P. L. Christiansen, J. C. Eilbeck, V. Z. Enolskii, and N. A. Kostov, \textit{Quasi-periodic solutions of the coupled nonlinear Schr\"{o}dinger equations }, Proc. Roy. Soc. London {\bf A 451} (1995), 685--700. \bibitem{13b} D. V. Chudnovsky, \textit{Meromorphic solutions of nonlinear partial differential equations and many-particle completely integrable systems}, J. Math. Phys. {\bf 20} (1979), 2416--2422. \bibitem{13e} D. V. Chudnovsky and G. V. Chudnovsky, \textit{Appendix I: Travaux de J. Drach (1919)}, Classical and Quantum Models and Arithmetic Problems (ed. by D. V. Chudnovsky and G. V. Chudnovsky), Marcel Dekker, New York, 1984, 445--453. \bibitem{13f} E. Colombo, G. P. Pirola, and E. Previato, \textit{Density of elliptic solitons}, J. reine angew. Math. {\bf 451} (1994), 161--169. \bibitem{13g} M. L. Da Silva Menezes, \textit{Infinite genus curves with hyperelliptic ends}, Commun. Pure Appl. Math. {\bf 42} (1989), 185--212. \bibitem{14} L. A. Dickey, \textit{Soliton Equations and Hamiltonian Systems}, World Scientific, Singapore, 1991. \bibitem{14a} R. Donagi and E. Markman, \textit{Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles}, Integrable Systems and Quantum Groups (ed. by R. Donagi, B. Dubrovin, E. Frenkel, and E. Previato), Springer, Berlin, 1996, 1--119. \bibitem{14b} J. Drach, \textit{D\'etermination des cas de r\'eduction de'l\'equation diff\'erentielle $d^2 y/dx^2=[\phi(x)+h]y$}, C. R. Acad. Sci. Paris {\bf 168} (1919), 47--50. \bibitem{14c} \bysame, \textit{Sur l'int\'egration par quadratures de'l\'equation $d^2 y/dx^2=[\phi(x)+h]y$ }, C. R. Acad. Sci. Paris {\bf 168} (1919), 337--340. \bibitem{15} B. A. Dubrovin, \textit{Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials}, Funct. Anal. Appl. {\bf 9} (1975), 215--223. \bibitem{15a} \bysame, \textit{Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties}, Funct. Anal. Appl. {\bf 11} (1977), 265--277. \bibitem{16} \bysame, \textit{Theta functions and non-linear equations}, Russ. Math. Surv. {\bf 36:2} (1981), 11--92. \bibitem{16a} \bysame, \textit{Matrix finite-zones operators}, Revs. Sci. Technology {\bf 23} (1983), 20--50. \bibitem{17} B. A. Dubrovin and S. P. Novikov, \textit{Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation}, Sov. Phys. JETP {\bf 40} (1975), 1058--1063. \bibitem{17a} J. C. Eilbeck and V. Z. Enol'skii, \textit{ Elliptic Baker-Akhiezer functions and an application to an integrable dynamical system}, J. Math. Phys. {\bf 35} (1994), 1192--1201. \bibitem{18} V. Z. Enol'skii, \textit{On the solutions in elliptic functions of integrable nonlinear equations}, Phys. Lett. {\bf 96A} (1983), 327--330. \bibitem{19} \bysame, \textit{On the two-gap Lam\'e potentials and elliptic solutions of the Kovalevskaja problem connected with them}, Phys. Lett. {\bf 100A} (1984), 463--466. \bibitem{20} \bysame, \textit{On solutions in elliptic functions of integrable nonlinear equations associated with two-zone Lam\'e potentials}, Soc. Math. Dokl. {\bf 30} (1984), 394--397. \bibitem{20b} V. Z. Enol'skii and J. C. Eilbeck, \textit{On the two-gap locus for the elliptic Calogero-Moser model}, J. Phys. {\bf A 28} (1995), 1069--1088. \bibitem{20a} V. Z. Enol'skii and N. A. Kostov, \textit{On the geometry of elliptic solitons}, Acta Appl. Math. {\bf 36} (1994), 57--86. \bibitem{fedor} M. V. Fedoryuk, \textit{Lam\'{e} wave functions in the Jacobi form}, J. Diff. Eqs., {\bf 23} (1987), 1170--1177. \bibitem{21} G. Floquet, \textit{Sur les \'equations diff\'erentielles lin\'eaires \`a coefficients doublement p\'eriodiques}, C. R. Acad. Sci. Paris {\bf 98} (1884), 38--39, 82--85. \bibitem{22} \bysame, \textit{Sur les \'equations diff\'erentielles lin\'eaires \`a coefficients doublement p\'eriodiques }, Ann. l'\'Ecole Normale Sup. {\bf 1} (1884), 181--238. \bibitem{23} \bysame, \textit{Addition a un m\'emorie sur les \'equations diff\'erentielles lin\'eaires}, Ann. l'\'Ecole Normale Sup. {\bf 1} (1884), 405--408. \bibitem{24} M. G. Gasymov, \textit{Spectral analysis of a class of second-order non-self-adjoint differential operators}, Funct. Anal. Appl. {\bf 14} (1980), 11--15. \bibitem{25} M. G. Gasymov, \textit{Spectral analysis of a class of ordinary differential operators with periodic coefficients}, Sov. Math. Dokl. {\bf 21} (1980), 718--721. \bibitem{25a} L. Gatto and S. Greco, \textit{Algebraic curves and differential equations: an introduction}, The Curves Seminar at Queen's, Vol. VIII (ed. by A. V. Geramita), Queen's Papers Pure Appl. Math. {\bf 88}, Queen's Univ., Kingston, Ontario, Canada, 1991, B1--B69. \bibitem{25b} I. M. Gel'fand and L. A. Dikii, \textit{Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations}, Russ. Math. Surv. {\bf 30:5}, (1975) 77--113. \bibitem{26} \bysame, \textit{Integrable nonlinear equations and the Liouville theorem }, Funct. Anal. Appl. {\bf 13} (1979), 6--15. \bibitem{27} F. Gesztesy, \textit{On the Modified Korteweg-deVries Equation}, Differential Equations with Applications in Biology, Physics, and Engineering (ed. by J. A. Goldstein, F. Kappel, and W. Schappacher), Marcel Dekker, New York, 1991, 139--183. \bibitem{27b} F. Gesztesy and R. Ratneseelan, \textit{An alternative approach to algebro-geometric solutions of the AKNS hierarchy}, preprint, 1996. \bibitem{27a} F. Gesztesy and W. Sticka, \textit{On a theorem of Picard}, preprint, 1996. \bibitem{28} F. Gesztesy and R. Weikard, \textit{Spectral deformations and soliton equations}, Differential Equations with Applications to Mathematical Physics (ed. by W. F. Ames, E. M. Harrell II, and J. V. Herod), Academic Press, Boston, 1993, 101--139. \bibitem{29} \bysame, \textit{Lam\'e potentials and the stationary (m)KdV hierarchy }, Math. Nachr. {\bf 176} (1995), 73--91. \bibitem{30} \bysame, \textit{Treibich-Verdier potentials and the stationary (m)KdV hierarchy}, Math. Z. {\bf 219} (1995), 451--476. \bibitem{31} \bysame, \textit{On Picard potentials}, Diff. Int. Eqs. {\bf 8} (1995), 1453--1476. \bibitem{32} \bysame, \textit{Floquet theory revisited}, Differential Equations and Mathematical Physics (ed. by I. Knowles), International Press, Boston, 1995, 67--84. \bibitem{32c} \bysame, \textit{A characterization of elliptic finite-gap potentials}, C. R. Acad. Sci. Paris {\bf 321} (1995), 837--841. \bibitem{32a} \bysame, \textit{Picard potentials and Hill's equation on a torus}, Acta Math. {\bf 176} (1996), 73--107. \bibitem{gray} J. Gray, \textit{Linear Differential Equations and Group Theory from Riemann to Poincar\'{e}}, Birkh\"{a}user, Boston, 1986. \bibitem{32b} S. Greco and E. Previato, \textit{Spectral curves and ruled surfaces: projective models}, The Curves Seminar at Queen's, Vol. VIII (ed. by A. V. Geramita), Queen's Papers Pure Appl. Math. {\bf 88}, Queen's Univ., Kingston, Ontario, Canada, 1991, F1--F33. \bibitem{33} V. Guillemin and A. Uribe, \textit{Hardy functions and the inverse spectral method}, Commun. PDE {\bf 8} (1983), 1455--1474. \bibitem{34} G.-H. Halphen, \textit{Trait\'e des Fonctions Elliptiques}, tome 2, Gauthier-Villars, Paris, 1888. \bibitem{35} C. Hermite, \textit{Oeuvres}, tome 3, Gauthier-Villars, Paris, 1912. \bibitem{36} E. L. Ince, \textit{Further investigations into the periodic Lam\'e functions}, Proc. Roy. Soc. Edinburgh {\bf 60} (1940), 83--99. \bibitem{37} E. L. Ince, \textit{Ordinary Differential Equations}, Dover, New York, 1956. \bibitem{38} A. R. Its and V. Z. Enol'skii, \textit{Dynamics of the Calogero-Moser system and the reduction of hyperelliptic integrals to elliptic integrals }, Funct. Anal. Appl. {\bf 20} (1986), 62--64. \bibitem{39} A. R. Its and V. B. Matveev, \textit{Schr\"odinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation}, Theoret. Math. Phys. {\bf 23} (1975), 343--355. \bibitem{39a} F. Klein, \textit{\"Uber den Hermite'schen Fall der Lam\'e'schen Differentialgleichung}, Math. Ann. {\bf 40} (1892), 125--129. \bibitem{40} N. A. Kostov and V. Z. Enol'skii, \textit{Spectral characteristics of elliptic solitons}, Math. Notes {\bf 53} (1993), 287--293. \bibitem{41} M. Krause, \textit{Theorie der doppeltperiodischen Funktionen einer ver\"anderlichen Gr\"osse}, Vol. 2, Teubner, Leipzig, 1897. \bibitem{42} I. M. Krichever, \textit{Integration of nonlinear equations by the methods of algebraic geometry}, Funct. Anal. Appl. {\bf 11} (1977), 12--26. \bibitem{43} \bysame, \textit{Methods of algebraic geometry in the theory of non-linear equations}, Russ. Math. Surv. {\bf 32:6} (1977), 185--213. \bibitem{44} \bysame, \textit{Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles}, Funct. Anal. Appl. {\bf 14} (1980), 282--290. \bibitem{44a} \bysame, \textit{Nonlinear equations and elliptic curves}, Revs. Sci. Technology {\bf 23} (1983), 51--90. \bibitem{44b} \bysame, \textit{Elliptic solutions of nonlinear integrable equations and related topics}, Acta Appl. Math. {\bf 36} (1994), 7--25. \bibitem{44c} V. B. Matveev and A. O. Smirnov, \textit{Symmetric reductions of the Riemann $\theta$--function and some of their applications to the Schr\"odinger and Boussinesq equation}, Amer. Math. Soc. Transl. (2) {\bf 157} (1993), 227--237. \bibitem{44d} H. P. McKean, \textit{Boussinesq's equation on the circle}, Commun. Pure Appl. Math. {\bf 34} (1981), 599--691. \bibitem{45} H. P. McKean and P. van Moerbeke, \textit{The spectrum of Hill's equation}, Invent. Math. {\bf 30} (1975), 217--274. \bibitem{45a} J. Mertsching, \textit{Quasi periodic solutions of the nonlinear Schr\"odinger equation}, Fortschr. Phys. {\bf85} (1987), 519--536. \bibitem{46} G. Mittag-Leffler, \textit{Sur les \'equations diff\'erentielles lin\'eaires \`a coefficients doublement p\'e\-rio\-di\-ques}, C. R. Acad. Sci. Paris {\bf 90} (1880), 299-300. \bibitem{46a} O. I. Mokhov, \textit{Commuting differential operators of rank 3, and nonlinear differential equations}, Math. USSR Izv., {\bf 35} (1990), 629--655. \bibitem{47} J. Moser, \textit{Integrable Hamiltonian systems and spectral theory}, Academia Nationale Dei Lincei, Scuola Normale Superiore, Lezione Fermiani, Pisa 1981, preprint, ETH-Z\"urich, 1982. \bibitem{47a} \bysame, \textit{Three integrable Hamiltonian systems connected with isospectral deformations}, Adv. Math. {\bf 16} (1975), 197--220. \bibitem{48} S. P. Novikov, \textit{The periodic problem for the Korteweg-de Vries equation}, Funct. Anal. Appl. {\bf 8} (1974), 236--246. \bibitem{49} S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, \textit{Theory of Solitons}, Consultants Bureau, New York, 1984. \bibitem{49b} M. Ohmiya, \textit{KdV polynomials and $\Lambda$--operator}, Osaka J. Math. {\bf 32} (1995), 409--430. \bibitem{49a} M. A. Olshanetsky and A. M. Perelomov, \textit{Classical integrable finite-dimensional systems related to Lie Algebras}, Phys. Rep. {\bf 71} (1981), 313--400. \bibitem{50} L. A. Pastur and V. A. Tkachenko, \textit{Spectral theory of Schr\"odinger operators with periodic complex-valued potentials}, Funct. Anal. Appl. {\bf 22} (1988), 156--158. \bibitem{51} L. A. Pastur and V. A. Tkachenko, \textit{An inverse problem for a class of one-dimensional Schr\"odinger operators with a complex periodic potential}, Math. USSR Izv. {\bf 37} (1991), 611--629. \bibitem{52} \bysame, \textit{Geometry of the spectrum of the one-dimensional Schr\"odinger equation with a periodic complex-valued potential}, Math. Notes {\bf 50} (1991), 1045--1050. \bibitem{52a} A. M. Perelomov, \textit{Integrable Systems of Classical Mechanics and Lie Algebras}, Vol.~1, Birkh\" auser, Basel, 1990. \bibitem{53} E. Picard, \textit{Sur une g\'en\'eralisation des fonctions p\'eriodiques et sur certaines \'equa\-tions diff\'eren\-tielles lin\'eaires}, C. R. Acad. Sci. Paris {\bf 89} (1879), 140--144. \bibitem{54} \bysame, \textit{Sur une classe d'\'equations diff\'erentielles lin\'eaires}, C. R. Acad. Sci. Paris {\bf 90} (1880), 128--131. \bibitem{55} \bysame, \textit{Sur les \'equations diff\'erentielles lin\'eaires \`a coefficients doublement p\'eriodiques}, J. reine angew. Math. {\bf 90} (1881), 281--302. \bibitem{55b} E. Previato, \textit{The Calogero-Moser-Krichever system and elliptic Boussineq solitons}, Hamiltonian Systems, Transformation Groups and Spectral Transform Methods (ed. by J. Harnard and J. E. Marsden), CRM, Montr\'eal, 1990, 57--67. \bibitem{55c} \bysame, \textit{Monodromy of Boussinesq elliptic operators}, Acta Appl. Math. {\bf 36} (1994), 49--55. \bibitem{55a} \bysame, \textit{Seventy years of spectral curves}, Integrable Systems and Quantum Groups (ed. by R. Donagi, B. Dubrovin, E. Frenkel, and E. Previato), Springer, Berlin, 1996, 419--481. \bibitem{56} F. S. Rofe-Beketov, \textit{The spectrum of non-selfadjoint differential operators with periodic coefficients}, Sov. Math. Dokl. {\bf 4} (1963), 1563--1566. \bibitem{56d} R. Schimming, \textit{An explicit expression for the Korteweg--de Vries hierarchy}, Acta Appl. Math. {\bf 39} (1995), 489--505. \bibitem{56c} I. Schur, \textit{\"Uber vertauschbare lineare Differentialausdr\"ucke}, Sitzungsber. Berliner Math. Gesellsch. {\bf 4} (1905), 2--8. \bibitem{57} G. Segal and G. Wilson, \textit{Loop groups and equations of KdV type}, Publ. Math. IHES {\bf 61} (1985), 5--65. \bibitem{58c} A. O. Smirnov, \textit{Real finite-gap regular solutions of the Kaup-Boussinesq equation}, Theoret. Math. Phys. {\bf 66} (1986), 19--31. \bibitem{58d} \bysame, \textit{A matrix analogue of Appell's theorem and reductions of multidimensional Riemann theta functions}, Math. USSR Sb. {\bf 61} (1988), 379--388. \bibitem{58} \bysame, \textit{Elliptic solutions of the Korteweg-de Vries equation}, Math. Notes {\bf 45} (1989), 476--481. \bibitem{58e} \bysame, \textit{Real elliptic solutions of the Sine-Gordon equation}, Math. USSR Sb. {\bf 70} (1990), 231--240. \bibitem{58a} \bysame, \textit{Finite-gap elliptic solutions of the KdV equation}, Acta Appl. Math. {\bf 36} (1994), 125--166. \bibitem{58b} \bysame, \textit{Solutions of the KdV equation elliptic in $t$}, Theoret. Math. Phys. {\bf 100} (1994), 937--947. \bibitem{58f} \bysame, \textit{Elliptic solutions of the nonlinear Schr\"odinger equation and the modified Korteweg-de Vries equation}, Russ. Acad. Sci. Sb. Math. {\bf 82} (1995), 461--470. \bibitem{59} I. A. Taimanov, \textit{Elliptic solutions of nonlinear equations}, Theoret. Math. Phys. {\bf 84} (1990), 700--706. \bibitem{59a} \bysame, \textit{On the two-gap elliptic potentials}, Acta Appl. Math. {\bf 36} (1994), 119--124. \bibitem{60} V. A. Tkachenko, \textit{Discriminants and generic spectra of non-selfadjoint Hill's operators}, BiBoS-preprint, Bochum, FRG, 1992. \bibitem{60b} A. Treibich, \textit{Compactified Jacobians of Tangential Covers}, Integrable Systems: The Verdier Memorial Conference (ed. by O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach), Birkh\"auser, Boston, 1993, 39--60. \bibitem{60a} \bysame, \textit{New elliptic potentials}, Acta Appl. Math. {\bf 36} (1994), 27--48. \bibitem{61} A. Treibich and J.-L. Verdier, \textit{Solitons elliptiques}, The Grothendieck Festschrift, Vol III (ed. by P. Cartier, L. Illusie, N. M. Katz, G. Laumon, Y. Manin, and K. A. Ribet), Birkh\"auser, Basel, 1990, 437--480. \bibitem{62} \bysame, \textit{Rev\^etements tangentiels et sommes de 4 nombres triangulaires}, C. R. Acad. Sci. Paris {\bf 311} (1994), 51--54. \bibitem{63} \bysame, \textit{Rev\^etements exceptionnels et sommes de 4 nombres triangulaires}, Duke Math. J. {\bf 68} (1992), 217--236. \bibitem{64} A. V. Turbiner, \textit{Lame equation, sl(2) algebra and isospectral deformations}, J. Phys. {\bf A22} (1989), L1--L3. \bibitem{65} J.-L. Verdier, \textit{New elliptic solitons}, Algebraic Analysis (ed. by M. Kashiwara and T. Kawai), Academic Press, Boston, 1988, 901--910. \bibitem{66} R. S. Ward, \textit{The Nahm equations, finite-gap potentials and Lam\'e functions}, J. Phys. {\bf A20} (1987), 2679--2683. \bibitem{66a} R. Weikard, \textit{Picard operators}, Math. Nachr., to appear. \bibitem{w2} \bysame, \textit{On second-order linear differential equations with inverse square singularities}, preprint, 1996. \bibitem{w3} \bysame, \textit{On Hill's equation with a singular complex-valued potential}, preprint, 1996. \bibitem{67} E. T. Whittaker and G. N. Watson, \textit{A Course of Modern Analysis}, Cambridge Univ. Press, Cambridge, 1986. \bibitem{68} G. Wilson, \textit{Commuting flows and conservation laws for Lax equations}, Math. Proc. Camb. Phil. Soc. {\bf 86} (1979), 131--143. \bibitem{69} G. Wilson, \textit{Algebraic curves and soliton equations}, Geometry Today (ed. by E. Arbarello, C. Procesi, and E. Strickland), Birkh\"auser, Boston, 1985, 303--329. \end{thebibliography} \end{document}