  An analytical solution of the perturbed equations is obtained, which exists
in all ergodic models of collisionless spherical stellar systems with a single
length parameter. This solution corresponds to variations of this parameter,
i.e., stretching or shrinking the sphere while preserving the total mass. The
system remains in an equilibrium state. The simplicity of the solution allows
for explicit expressions for the distribution function, potential, and density
at all orders of perturbation theory. This, in turn, helps to clarify the
concept of perturbation energy, which, being a second-order quantity in
amplitude, cannot be calculated in linear theory. It is shown that the correct
expression for perturbation energy, constructed taking into account 2nd order
perturbations, and the well-known expression for perturbation energy
constructed as bilinear form obtained within linear theory from 1st order
perturbations do not coincide. However, both of these energies are integrals of
motion and differ only by a constant. The obtained solution can be used to
control the correctness of codes and the accuracy of calculations in the
numerical study of collisionless stellar models.
