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A shape invariant nonseparable and nondiagonalizable two-dimensional modelwith quadratic complex interaction, first studied by Cannata, Ioffe, andNishnianidze, is re-examined with the purpose of exhibiting its hiddenalgebraic structure. The two operators $A^+$ and $A^-$, coming from the shapeinvariant supersymmetrical approach, where $A^+$ acts as a raising operatorwhile $A^-$ annihilates all wavefunctions, are completed by introducing a novelpair of operators $B^+$ and $B^-$, where $B^-$ acts as the missing loweringoperator. These four operators then serve as building blocks for constructing${\mathfrak{gl}}(2)$ generators, acting within the set of associated functionsbelonging to the Jordan block corresponding to a given energy eigenvalue. Thisanalysis is extended to the set of Jordan blocks by constructing two pairs ofbosonic operators, finally yielding an ${\mathfrak{sp}}(4)$ algebra, as well asan ${\mathfrak{osp}}(1/4)$ superalgebra. Hence, the hidden algebraic structureof the model is very similar to that known for the two-dimensional realharmonic oscillator.

symmetry, integrability and geometry: methods and applications

Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable  and Nondiagonalizable Modelswith Quadratic Complex Interaction. I.  Two-Dimensional Model