  The problem of d-dimensional Schrodinger equations with a position-dependent
mass is analyzed in the framework of first-order intertwining operators. With
the pair (H, H_1) of intertwined Hamiltonians one can associate another pair of
second-order partial differential operators (R, R_1), related to the same
intertwining operator and such that H (resp. H_1) commutes with R (resp. R_1).
This property is interpreted in superalgebraic terms in the context of
supersymmetric quantum mechanics (SUSYQM). In the two-dimensional case, a
solution to the resulting system of partial differential equations is obtained
and used to build a physically-relevant model depicting a particle moving in a
semi-infinite layer. Such a model is solved by employing either the
commutativity of H with some second-order partial differential operator L and
the resulting separability of the Schrodinger equation or that of H and R
together with SUSYQM and shape-invariance techniques. The relation between both
approaches is also studied.
