  In this paper we investigate existence and characterization of non-radial
pseudo-radial (or separable) solutions of some semi-linear elliptic equations
on symmetric 2-dimensional domains. The problem reduces to the phase plane
analysis of a dynamical system. In particular, we give a full description of
the set of pseudo-radial solutions of equations of the form $\Delta u = \pm
a^2(|x|) u|u|^{q-1}$, with $q>0$, $q\neq 1$. We also study such equations over
spherical or hyperbolic symmetric domains.
