  We study the RPA equations in their most general form by taking the matrix
elements appearing in the RPA equations as random. This yields either a
unitarily or an orthogonally invariant random-matrix model which is not of the
Cartan type. The average spectrum of the model is studied with the help of a
generalized Pastur equation. Two independent parameters govern the behaviour of
the system: The strength $\alpha^2$ of the coupling between positive- and
negative-energy states and the distance between the origin and the centers of
the two semicircles that describe the average spectrum for $\alpha^2 = 0$, the
latter measured in units of the equal radii of the two semicircles. With
increasing $\alpha^2$, positive- and negative-energy states become mixed and
ever more of the spectral strength of the positive-energy states is transferred
to those at negative energy, and vice versa. The two semicircles are deformed
and pulled toward each other. As they begin to overlap, the RPA equations yield
non--real eigenvalues: The system becomes unstable. We determine analytically
the critical value of the strength for the instability to occur. Several
features of the model are illustrated numerically.
