Two-dimensional supersymmetry: From SUSY quantum mechanics to integrable classical models

Two known two-dimensional SUSY quantum mechanical constructions-the direct generalization of SUSY with first-order supercharges and higher-order SUSY with second-order supercharges-are combined for a class of 2-dim quantum models, which are not amenable to separation of variables. The appropriate classical limit of quantum systems allows us to construct SUSY-extensions of original classical scalar Hamiltonians. Special emphasis is placed on the symmetry properties of the models thus obtained-the explicit expressions of quantum symmetry operators and of classical integrals of motion are given for all (scalar and matrix) components of SUSY-extensions. Using Grassmanian variables, the symmetry operators and classical integrals of motion are written in a unique form for the whole Superhamiltonian. The links of the approach to the classical Hamilton-Jacobi method for related “flipped” potentials are established.