  We study a quantum model with non-isotropic two-dimensional oscillator
potential but with additional quadratic interaction $x_1x_2$ with imaginary
coupling constant. It is shown, that for a specific connection between coupling
constant and oscillator frequences, the model {\it is not} amenable to a
conventional separation of variables. The property of shape invariance allows
to find analytically all eigenfunctions and the spectrum is found to be
equidistant. It is shown that the Hamiltonian is non-diagonalizable, and the
resolution of the identity must include also the corresponding associated
functions. These functions are constructed explicitly, and their properties are
investigated. The problem of $R-$separation of variables in two-dimensional
systems is discussed.
