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JOURNAL OF MATHEMATICAL PHYSICS, vol.42, no.8, pp.3344-3360, 2001 (SCI-Expanded)
The method of intertwining with n-dimensional (nD) linear intertwining operator L is used to construct nD isospectral, stationary potentials. It has been proven that the differential part of L is a series in Euclidean algebra generators. Integrability conditions of the consistency equations are investigated and the general form of a class of potentials respecting all these conditions have been specified for each n=2, 3, 4, 5. The most general forms of 2D and 3D isospectral potentials are considered in detail and construction of their hierarchies is exhibited. The followed approach provides coordinate systems which make it possible to perform separation of variables and to apply the known methods of supersymmetric quantum mechanics for 1D systems. It has been shown that in choice of coordinates and L there are a number of alternatives increasing with n that enlarge the set of available potentials. Some salient features of higher dimensional extension as well as some applications of the results are presented. (C) 2001 American Institute of Physics.