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{\nopagenumbers
~ \vskip 2 truecm
\centerline{\bf Commuting-flow symmetries and common solutions}
\medskip
\centerline{\bf to differential equations with common symmetries}
\vskip 3 truecm
\centerline{Giuseppe Gaeta$^*$}
\medskip
\centerline{\it I.H.E.S., 35 Route de Chartres}
\centerline{\it 91440 Bures sur Yvette (France)}
\centerline{\tt gaeta@ihes.fr}
\bigskip
\bigskip
\centerline{Paola Morando}
\medskip
\centerline{\it Dipartimento di Matematica, Politecnico di Torino,}
\centerline{\it Corso Duca degli Abruzzi 24, 10129 Torino (Italy)}
\centerline{\tt morando@polito.it}
\vskip 4 truecm
{\bf Abstract:} We point out that in certain cases, all the differential
equations (for given indipendent and dependent variables) possessing a
given symmetry necessarily share a common solution. We characterize this
situation and the common solution, and give concrete examples.
\vfill \parindent=0pt
{\petitrm $^*$
Permanent address: Dept. of Mathematics,
Loughborough University, Loughborough LE11 3TU (Great Britain).}
\footnote{}{{\tt \giorno}}
\eject}
\pageno=1
\parskip=10pt
\parindent=0pt
~ \bigskip\bigskip
{\bf Introduction}
\bigskip
We are all familiar with elementary occurences of a remarkable
phenomenon: in some cases, {\it all} the differential equations
possessing a given symmetry share a common solution.
As trivial examples of this fact it suffices to think of reflection-invariant
ODEs in $\R^1$, ${\dot x} = f(x,t)$ [these are identified by $f(x,t) = -
f(-x,t)$], which all share the common solution $x(t) \equiv 0$; or also
reflection and/or rotation invariant ODEs in $\R^n$, which again share
the common (trivial) solution $x(t) \equiv 0$.
The purpose of this short note is to point out that this phenomenon can
be present in far less trivial cases, for both ODEs and PDEs; and to
give a way to characterize the common solutions, at least for a
relevant class of symmetries, i.e. the ``commuting-flow symmetries''
defined below (which are natural in a dynamical systems geometric
approach).
This question can be tackled making use of Michel's theory (see below),
and indeed our result can be seen as a simple corollary -- maybe
reaching unexpected fields -- of the original Michel theorem.
A classical result of L. Michel \ref{1} ensures that, given a
finite dimensional smooth manifold $M$ with an action of a compact Lie
group $G$, we can identify the points $x \in M$ which are critical for
{\it all} the $G$-invariant potentials $V(x)$. The investigation of
Michel was motivated principally by fundamental problems in Physics (the
theory of strong interactions) \ref{2}, but the theorem in itself is of
a geometric nature, and of a much wider applicability to a range of
problems involving {\it spontaneous symmetry breaking}.
The Michel's result can be suitably generalized to variational problems
in function spaces \ref{3}, and it has been recently extended to
consider gauge invariant functionals \ref{4}.
On the other side, the proof given by Michel in \ref{1} for his
theorem immediately generalizes to consider equivariant dynamics
defined by a vector field on $M$, and its fixed points; it also
guarantees, more generally, the invariance under the equivariant
dynamics of a family of subspaces of $M$ (identified by sharing a
common isotropy subgroup for the $G$-action). In this way, one could
also readily recover a number of fundamental results in equivariant
dynamics and equivariant bifurcation theory, such as the reduction
lemma of Golubitsky and Stewart \ref{5} and the equivariant branching
lemma of Cicogna and Vanderbauwhede \ref{6}, originally obtained with
no use of Michel's theory \ref{7}.
It is thus quite natural to ask if, beside the extension of the
variational aspects of Michel theory to infinite dimensional
spaces \ref{4}, one
could not obtain also a similar extension for its equivariant dynamics
aspects. The purpose of this note is to show that this is indeed the
case, and namely that we can identify subsets of a given function space
$\F$ which are invariant under any dynamics on $\F$ which satisfy
certain symmetry requirements with respect to an algebra $\G$ of vector
fields; moreover, proceeding pretty much as in the finite dimensional
case, we can identify functions $f_0 \in \F$ which are solutions for
{\it any} differential equation $\De$ which is $\G$-invariant (in a
sense to be precisely defined in the following).
Our discussion makes use of ideas and formalisms from the symmetry
theory of differential equations \ref{8-11}, which we
assume is known -- at least in its basics -- to the reader. Also, our
discussion will be conducted at a rather formal level; thus, in
particular, we will assume the existence of flows generated by the
(generalized) vector fields to be considered in the following.
\vfill\eject
{\bf 1. Evolutionary vector fields and symmetry}
\bigskip
Let us consider a smooth manifold $M = \X \times \Y$, where the $x
\in \X \sse \R^q$ and the $u \in \Y \sse \R^p$ will be thought as,
respectively, the independent and dependent variables for the
differential equations to be considered in a moment.
We consider a vector field $Y$ on $M$, which we write as
$$ Y \ = \ \sum_{i=1}^q \xi^i (x,u) {\pa \over \pa x^i} \ + \ \sum_{\a=1}^p
\phi^\a (x,u) {\pa \over \pa u^\a} \ \eqno(1) $$
(with $\xi$, $\phi$ smooth functions on $M$); as it is well known
\ref{8}, to any such vector field we can associate an {\it
evolutionary representative}; this will be denoted by $X$ and is given by
$$ X \ = \ \[ \phi^\a (x,u) - \sum_i \xi^i (x,u) u^\a_i \] \pa_\a \
\equiv Q^\a [u] \pa_\a \ . \eqno(2) $$
Here we write $\pa_\a $ for $\pa / \pa u^\a$, and $u^\a_i$ for $\pa u^\a
/ \pa x^i$ (these notations will be understood throughout in the
following; from now on we will also understand summation over repeated
indices).
The notation $Q[u]$ -- used extensively in
the following -- means
that $Q$ is a function of $x$, $u$, and $u$ derivatives of any order
(although in the present case these are only of first order).
We will also write $X$ as $X_Q$, to stress the association with the
vector $Q[u]$; more generally, $X_P$ will denote the evolutionary
vector field $X_P = P^\a [u] \pa_\a$.
It should be recalled that, while $Y$ is a legitimate vector field on
$M$, its evolutionary representative $X$ should be considered with some
extra care; we can either consider it as a notation for a vector field in
the function space $\M$ (the space of smooth functions from $\R^q$ to
$\R^p$), such that $ \( I + \eps X_Q \) $ transforms $f(x)$ into the
function
$$ \~f^\a (x) = f^\a (x) + \eps \[ \phi^\a (x, f(x) ) - \xi^i
(x,f(x)) f^\a_i (x) \] \ , \eqno(3) $$
either as a generalized vector field acting in the jet space associated
to $M$; in this case, one should more precisely consider the
infinite order prolongation $X^*$ of $X$, acting in the infinite order
jet space $J^* (M)$, in order to have a proper vector
field.
It should also be stressed that, for the argument to be considered below,
one could start directly from $X^*$, i.e. consider generalized vector
fields rather than limit to consider only representative of geometric
vector fields.
Let us now consider a system of $p$ differential equations for functions
$u^\a = u^\a (x) \in \M$; this will be written as $\De [u] = 0$ (this
will be a vector $\De^\a$, with $\a = 1,...,p$; we will say for
simplicity ``the equation'' $\De$ to refer to this system). With the
equation $\De$ we associate an evolutionary generalized vector field
$X_\De$ given by
$$ X_\De \ = \ \De^\a [u] \pa_\a \ ; \eqno(4) $$
as it was the case for $X_Q$, this can be seen either as a vector field on
$\M$, either as denoting the infinite prolongation $X_\De^*$ acting in
$J^* (M)$.
We recall \ref{8} that $X_Q$ is a {\it symmetry} of $\De$ if
$$ X_Q^* (\De ) \ = \ 0 \ , \eqno(5) $$
is satisfied whenever $\De = 0$,
and a {\it strong symmetry} of $\De$ if (5) is identically satisfied.
\bigskip\bigskip
\vfill\eject
{\bf 2. \boh symmetries}
\bigskip
Given two evolutionary generalized vector fields (EGVF), $X_P = P^\a[u]
\pa_\a$ and $X_Q = Q^\a [u] \pa_\a$, we can consider their commutator
$\[ X_P , X_Q \]$; this will be again an EGVF $Y_R = R^\a [u] \pa_\a$,
with
$$ R^\a [u] = X_P^* \( Q^\a \) - X_Q^* \( P^\a \) \ . \eqno(6) $$
It is easy to check \ref{8} that this gives indeed a proper
commutator, i.e. the usual properties of the commutator are satisfied.
Given a differential equation $\De$, and an EGVF $X_Q$, we say that
$X_Q$ is a {\bf Commuting Flow (CF) symmetry} for $\De$ if the
associated EGVF commute (in the above sense), i.e. if
$$ \[ X_Q , X_\De \] \ = \ 0 \ . \eqno(7) $$
For a given equation $\De$, we denote by $\G_\De$ the set of EGVF which
commute with $X_\De$ (this is easily seen to be a Lie algebra, with $X_\De$
belonging to its center).
Conversely, for a given EGVF $X_Q $ we denote by $\I_Q$ the set of
differential equations which have $X_Q$ as a \boh symmetry; again, the
$X_\De$ corresponding to $\De \in \I_Q$ form a Lie algebra.
{\bf Lemma 1.} {\it If $X_Q$ is a \boh symmetry for $\De$, then it is
also a symmetry of $\De$ in the ordinary sense.}
{\tt Proof.} Indeed, on $\De =0$ we have that both $X_\De$ and
$X_\De^*$ vanish, as it is clear since $X_\De = \De^\a [u] \pa_\a$ and
$X_\De^* = \sum_J (D_J \De^\a [u] ) (\pa / \pa u^\a_J )$ (here $J$ is
again a multiindex). Thus, $X_\De^* (Q) = 0$ on $\De=0$ for any
$Q$, and the vanishing of $\[ X_Q , X_\De \] $ is equivalent to the
vanishing of $X_Q^* ( \De )$, which is the condition (4) for $X_Q$ to be
a symmetry of $\De$ in the ordinary sense. $\odot$
\bigskip\bigskip
{\bf 3. Invariance of fixed space and common solutions to symmetric
equations}
\bigskip
With any $X_Q : \F \to \T \F$ we associate the set $F(Q) \sse \F$ of
functions which are fixed points for the vector field $X_Q$;
equivalently, this means that $Q[f] = 0$ for $f \in F(Q)$.
{\bf Lemma 2.} {\it If $X_Q$ is a \boh symmetry of $\De$, then $X_\De$
leaves $F(Q)$ globally invariant, i.e. $X_\De : F(Q) \to \T F(Q)$.}
{\tt Proof.} Let us consider, for the sake of clarity, the integrated
version of the commutation relations (assuming suitable existence and
unicity conditions for the flows of the vector fields);
we have -- with $\la$, $\mu$ real parameters
-- that $$ e^{\la Q[u]} e^{\mu \De [u]} (f) \ = \ e^{\mu \De [u]} e^{\la
Q[u]} (f) \ ; \eqno(8) $$
from $f \in F(Q)$ it follows that $e^{\la Q} (f) = f$, and thus we have
that $$ e^{\mu \De [u] } \( f \) \in F(Q) \quad \forall \mu \ ,
\eqno(9) $$ from which the lemma follows immediately. $\odot$
{\bf Corollary.} {\it If there exists an $f_0 \in F(Q)$ isolated in
$F(Q)$, then necessarily $f_0$ is solution to any $\De \in \I_Q$.}
{\tt Proof.} In this case, $f_0$ is a fixed point of $\De [u]$; but
this is equivalent to being a solution to $\De$. $\odot$
\remark When we speak of an ``isolated'' function, we are implicitely
assuming $\M$ is equipped with some natural topology; in the examples
below, in which we consider a space of $L^2$ functions, this will be the
$L^2$ norm. Actually, we could consider any topology in which the flows
of $X_Q$ and $X_\De$ are continuous.
The above lemma and corollary are immediately generalized to the case
of a whole algebra of \boh symmetries; we denote by $\G \equiv \G_\De$
this (possibly,
but not necessarily, abelian) algebra, and say that $\G$ is a \boh
symmetry algebra for $\De$ if $[X_Q , X_\De ] = 0$ for any $X_Q \in
\G$. In this case we denote by $F (\G )$ the
set of functions invariant under all the $X_Q \in \G$, and by $\I_\G$
the set of equations for which $\G$ is a \boh symmetry algebra. We have
then immediately the
{\bf Lemma 3.} {\it If $\G$ is a \boh symmetry algebra of $\De$, then
$X_\De$ leaves $F(\G )$ globally invariant, i.e. $X_\De : F(\G ) \to \T
F(\G )$.}
{\bf Corollary.} {\it If there exists a $f_0 \in F(\G )$ isolated in
$F(\G )$, then necessarily $f_0$ is solution of any $\De \in \I_\G$.}
It should be noted that it is not easy to have such an isolated $f_0 \in
F(Q) \sse \M$; e.g., in the case where $Q[u]$ is a homogeneous linear
function of $u$,
we have that necessarily $F(Q)$ is a linear subspace of $\M$, and we
cannot have isolated points (unless $F[Q]$ reduces to the zero function
alone).
However, we have so far considered the whole function space $\M$; when
we deal with differential equations, we usually have to look
for solutions in some given function space, less general than $\M$;
restricting to a smaller function space makes it easier to find isolated
fixed points $f_0$.
Let us consider a closed subspace $\F \sse \M$, which we think as fixed
once and for all. We have now to consider only those $\De$
such that $X_\De$ are vector fields on $\F$. We can then
obtain again the equivalent of the lemma 3 and its corollary given above,
provided we consider now the intersection of $F(\G ) \sse \M$ with $\F$.
We summarize our discussion in the following form.
{\bf Proposition.} {\it Let $\G$ be an algebra of EGVF on $\M$, and $F(\G
)$ the space of fixed points for $\G$; let $\F \sse \M$ be a subspace of
$\M$, such that there exists a $f_0$ isolated in $F(\G ) \cap \F$.
Then, any differential equation $\De \in \I_\G$ such that $X_\De : \F
\to \T \F$ admits $f_0 (x) \in \F$ as a solution.}
\bigskip \bigskip
{\bf 4. Examples}
\bigskip
{\it Example 1.}
Let us consider $\X = S^1$, $\Y = R$, so that $\M$ is the space of
$\C^\infty$ function $f:S^1 \to R$; in this we select the space $\F$ of
functions which satisfy $$ ||f||^2 \ \equiv \ {1 \over 2 \pi}
\int_{0}^{2 \pi} | f(x) |^2 \d x \ = \ 1 \ . \eqno(10) $$
We consider $Y = - \pa / \pa x$, and correspondingly
$$ X_Q = u_x \pa_u \quad ; \quad Q[u] = u_x \eqno(11) $$
Let us now look for $F(Q) \cap \F$; $F(Q)$ is given by constant
functions $f(x) = c$, and the constraint (10) selects the two functions
$ f_\pm (x) = \pm 1 $. These are obviously isolated in $F(Q) \cap \F$.
Let us now determine the $\De \in \I_Q$; the condition $\[ X_Q , X_\De
\] = 0$ amounts to $\pa \De / \pa x = 0$. Indeed, $X_Q^* = \sum_j (D_j
u_x ) \pa_{u_j}$, while the first prolongation of $X_\De$ is given by
$X_\De^{(1)} = \De \pa_u + (D_x \De ) \pa_{u_x}$. Thus,
$$ X_Q^* (\De ) - X_\De^* (Q) = \sum_j (D_j u_x ) {\pa \De \over \pa u_j} -
D_x \De \ . \eqno(12) $$
Expanding $D_x \De = \De_x + \sum_j (\pa \De / \pa u_j ) u_{x,j}$ (where
$u_{x,j} = D_j u_x$), we get that (12) reduces to $\De_x$.
Thus, $\De \in \I_Q$ are written as
$$ \De \ = \ \De \( u, u_x , u_{xx}, ... \) = 0 $$
Now we have to check that all such $\De$ which moreover have $X_\De$ leaving
$\F$ invariant admit $f_\pm (x)$ as solutions.
The requirement $X_\De : \F \to \F$ amounts to asking that $$ || e^{\la X_\De}
(f) || = 1 \eqno(13) $$ whenever $|| f|| = 1$. In the limit $\la \to
0^+$, $$ e^{\eps X_\De} (f) \ \simeq \ \( I + \eps X_\De \) (f) \simeq f
+ \eps \De [f] \ , $$
and thus
$$ || e^{\la X_\De} (f) ||^2 = || f ||^2 + 2 \eps \< f \cdot \De [f] \> + o
(\eps ) \ . \eqno(14) $$
As (13) must hold for any $f \in \F$, it must in particular hold on $f_\pm$;
evaluating (14) in $f_\pm$ we obtain then $\De ( \pm 1 , 0 , 0 , 0 , ... ) = 0$, i.e.
that $f_\pm$ is a solution to $\De$.
\bigskip
{\it Example 2.}
We consider now an example with $u=u(x_1,x_2)$ and an algebra of \boh
symmetries spanned by two vector fields, i.e.
$$ \eqalign{
Y_1 \ =& \ - x_1 \pa_{x_2} + x_2 \pa_{x_1} = - \pa_\theta \cr
Y_2 \ =& \ x_1 \pa_{x_1} + x_2 \pa_{x_2} + k u \pa_u = r \pa_r + ku
\pa_u \ , \cr} \eqno(15) $$
where $(r,\theta )$ are polar coordinates in the $(x_1,x_2)$ plane;
clearly, $Y_1$ represents a rotation in the independent variables, and
$Y_2$ a scale transformation.
Correspondingly, we have $$ X_1 = u_\theta
\pa_u \quad , \quad X_2 = (k u - u u_r ) \pa_u \ . \eqno(16) $$
and it is easy to see that
$$ F (Q_1 ) = \{ u=u(r) \} \quad , \quad F (Q_2 ) = \{ u(x_1 , x_2 ) =
\sum_{b=0}^k c_b x_1^b x_2^{k-b} \} \eqno(17) $$
(with $c_b$ real constants) and therefore
$$ F(\G ) \ = \ \{ u = \a r^k \} \qquad (\a \in \R ) \ . \eqno(18) $$
We will consider, for the sake of clarity, only first order
polynomial PDEs, i.e. we assume
$$ \De \ = \ \De \( r , \theta ; u , u_r , u_\theta \) \ ; \eqno(19)
$$ this also means that
$$ X_\De^{(1)} \ = \ \De \pa_u + \( D_r \De \) \pa_{u_r} + \( D_\theta
\De \) \pa{u_\theta} \eqno(20) $$
(in the following computations, we only need this first order
prolongation of $X_\De$). With this, it follows from straightforward
computations that
$$ \eqalignno{
\[ X_1 , X_\De \] \ =& \ - {\pa \De \over \pa \theta} & (21) \cr
\[ X_2 , X_\De \] \ =& \ - k \De + r {\pa \De \over \pa r} + k u {\pa
\De \over \pa u} + k u_\theta {\pa \De \over \pa u_\theta} + (k-1) u_r
{\pa \De \over \pa u_r} & (22) \cr} $$
The (21), and the polynomial assumption on $\De$, guarantees that $\De
\in \I_{Q_1}$ implies
$$ \De \ = \ \ga_{abcd} r^a u^b u_r^c u_\theta^d \ ;
\eqno(23) $$
moreover, (22) requires that, in order to have $\De \in \I_\G$, only
the $\ga_{abcd}$ with
$$ k \ (b+c+d-1) \ = \ c-a \eqno(24) $$
can be nonzero.
We can now check that indeed $X_\De$ leaves the space of functions
of the form $u = \a r^k $ invariant. In facts,
$$ X_\De (\a r^k ) \ = \ \ga_{abcd} r^a \a^b r^{kb} (\a k)^c r^{c(k-1)}
\delta_{d,0} \ = \ \b \ga_{abc0} r^{a + kb + c(k-1)} \ , \eqno(25)
$$ and it follows from (24) that the exponent of $r$ in the final
expression is just $k$. This shows that indeed $X_\De : F(\G ) \to \T F(\G )$,
as claimed.
Let us now consider again $\F$ given by the functions of unit norm,
where we define
$$ || f ||^2 = \< f , f \> \equiv \int_D | f(x_1 , x_2 ) |^2 \d x_1 \d
x_2 \eqno(26) $$ (here $D$ is the unit disk $r \le 1$); in this case
the only functions of $F(\G )$ which are also in $\F$ are
$$ f_\pm (x_1 , x_2 ) = \pm \a_0 r^k \qquad [\a_0 = \sqrt{(k+1)/\pi} ] \ .
\eqno(27) $$
We rewrite the action of $X_\De$ as
$$ \( I + \eps X_\De \) (f) = f + \eps \De [f] \ ; \eqno(28) $$
thus -- for $||f||=1$ -- the norm of this is one, i.e. $X_\De $ leaves
$\F$ invariant, if and only if $$ \< f , \De [f] \> \ = \ 0 \ .
\eqno(29) $$
We notice that $F(\G )$ is a one dimensional linear space, so that
$X_\De : F(\G ) \to \T F(\G )$ means that
necessarily $X_\De (f) = \la f$, for some real number $\la$, for any $f
\in F( \G )$. With this, (29) means
$$ \< f , \la f \> \equiv \la ||f||^2 = 0 \ , \eqno(30) $$
which can be true only for $\la = 0$. This shows that the condition
that $X_\De : \F \to \T \F$ guarantees also that the isolated points in
$F(\G ) \cap \F$ are solutions to $\De$, as claimed by our theorem.
\bigskip
{\it Example 3.}
Finally, let us consider $\X = R \times R_+$ and $\Y = R^1$; then $\M$ is the space
of $\C^\infty$ functions $f : R^2 \to R$, and in this we select the space
$\F$ of functions which satisfy, with $(x,t)$ coordinates in $\X$,
$$ \int_{- \infty}^{+ \infty} f(x,t) \d x \ = \ 1 \ \eqno(31) $$
and such that $f_x(x,t)$ go to zero when $x$ go to infinity.
We consider $Y = 2 t \pa_x - xu \pa_u$, and correspondingly
$$ X \ = \ - \( xu + 2 t u_x \) \pa_u \ ;\eqno(32) $$
notice that $Y$ is one of the Lie-point symmetries of the heat equation
\ref{8-11}, and it can be checked that $X$ is a \boh symmetry of it.
It can also be checked easily that the functions in $F(Q)$
are those of the form
$$ f(x,t) \ = \ k(t) \ e^{- x^2 /4 t} \ , \eqno(33) $$
and the intersection of this $F(Q)$ with $\F$ yields precisely
$$ f_0 (x,t) \ = \ {1 \over \sqrt{4 \pi t} } e^{- x^2 / 4t } \ , \eqno(34) $$
i.e. the fundamental solution of the heat equation.
\vfill\eject
~
\bigskip\bigskip
{\bf References}
\bigskip\bigskip
\def\CMP{{\petitit Comm. Math. Phys.}}
\def\ref#1{\item{[#1]} }
\baselineskip=10pt
\def\tit#1{{``#1''}}
\ref{1} L. Michel, \tit{Points critiques de fonctions invariantes sur
une G-vari\'et\'e}, {\it Comptes Rendus Acad. Sci. Paris} {\petitbf
272-A} (1971), 433-436
\ref{2} L. Michel and L. Radicati, \tit{Properties of the breaking of
hadronic internal symmetry}, {\it Ann. Phys. (N.Y.)} {\petitbf 66}
(1971), 758; \tit{The geometry of the octet}, {\it Ann. I.H.P.}
{\petitbf 18} (1973), 185
\ref{3} R.S. Palais, \tit{The principle of symmetric criticality},
{\it Comm. Math. Phys.} {\petitbf 69} (1979), 19-30
\ref{4} G. Gaeta and P. Morando, \tit{Michel theory of symmetry
breaking and pure gauge theories}, preprint 1996
\ref{5} M. Golubitsky and I.N. Stewart, \tit{Hopf bifurcation in the
presence of symmetry}, {\it Arch. Rat. Mech. Anal.} {\petitbf 87}
(1985), 107-165; M. Golubitsky, D. Schaeffer and I. Stewart, {\it
Singularities and groups in bifurcation theory - vol. II}, Springer
(New York) 1988
\ref{6} G. Cicogna, \tit{Symmetry breakdown from bifurcation},
{\petitit Lett. Nuovo Cimento} {\bf 31} (1981), 600-602; and \tit{A
nonlinear version of the equivariant bifurcation lemma}, {\petitit J.
Phys. A} {\bf 23} (1990), L1339-L1343; A. Vanderbauwhede, {\petitit
Local bifurcation and symmetry}, Pitman (Boston) 1982
\ref{7} G.Gaeta, \tit{A splitting lemma for equivariant dynamics},
{\it Lett. Math. Phys.} {\petitbf 33} (1995), 313-320;
\tit{Splitting equivariant dynamics}, {\it Nuovo Cimento B}
{\petitbf 110} (1995), 1213-1226
\ref{8} P.J. Olver, {\it Applications of Lie groups to differential
equations}, Springer (New York) 1986
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\ref{10} H. Stephani, {\it Differential equations. Their solution
using symmetries}, Cambridge University Press (Cambridge UK) 1989
\ref{11} G. Gaeta, {\petitit Nonlinear symmetries and nonlinear
equations}, Kluwer (Dordrecht) 1994
\bye