Mapping of Two-Dimensional Schr\"odinger Equation under the Point Transformation

For the two-dimensional Schr"odinger equation, the general form of the point transformations such that the result can be interpreted as a Schr"odinger equation with effective (i.e. position dependent) mass is studied. A wide class of such models with different forms of mass function is obtained in this way. Starting from the solvable two-dimensional model, the variety of solvable partner models with effective mass can be built. Several illustrating examples not amenable to the conventional separation of variables are given.