  Two new methods for investigation of two-dimensional quantum systems, whose
Hamiltonians are not amenable to separation of variables, are proposed. 1)The
first one - $SUSY-$ separation of variables - is based on the intertwining
relations of Higher order SUSY Quantum Mechanics (HSUSY QM) with supercharges
allowing for separation of variables. 2)The second one is a generalization of
shape invariance. While in one dimension shape invariance allows to solve
algebraically a class of (exactly solvable) quantum problems, its
generalization to higher dimensions has not been yet explored. Here we provide
a formal framework in HSUSY QM for two-dimensional quantum mechanical systems
for which shape invariance holds. Given the knowledge of one eigenvalue and
eigenfunction, shape invariance allows to construct a chain of new
eigenfunctions and eigenvalues. These methods are applied to a two-dimensional
quantum system, and partial explicit solvability is achieved in the sense that
only part of the spectrum is found analytically and a limited set of
eigenfunctions is constructed explicitly.
