  The so-called ``symplectic method'' is used for studying the linear stability
of a self-gravitating collisionless stellar system, in which the particles are
also submitted to an external potential. The system is steady and spherically
symmetric, and its distribution function $f_0$ thus depends only on the energy
$E$ and the squarred angular momentum $L^2$ of a particle. Assuming that
$\partial f_0 / \partial E < 0$, it is first shown that stability holds with
respect to all the spherical perturbations -- a statement which turns out to be
also valid for a rotating spherical system. Thus it is proven that the energy
of an arbitrary aspherical perturbation associated to a ``preserving generator"
$\delta g_1$ [i.e., one satisfying $\partial f_0 / \partial L^2 \{ \delta g_1,
L^2 \} = 0$] is always positive if $\partial f_0 / \partial L^2 \leq 0$ and the
external mass density is a decreasing function of the distance $r$ to the
center. This implies in particular (under the latter condition) the stability
of an isotropic system with respect to all the perturbations. Some new remarks
on the relation between the symmetry of the system and the form of $f_0$ are
also reported. It is argued in particular that a system with a distribution
function of the form $f_0 = f_0 (E,L^2)$ is necessarily spherically symmetric.
