Hidden nonlinear su(2|2) superunitary symmetry of N=2 superextended 1D Dirac delta potential problem

We show that the N=2 superextended 1D quantum Dirac delta potential problem is characterized by the hidden nonlinear $su(2\mathrm{\mid }2)$ superunitary symmetry. The unexpected feature of this simple supersymmetric system is that it admits three different ${\mathbb{Z}}_2$-gradings, which produce a separation of 16 integrals of motion into three different sets of 8 bosonic and 8 fermionic operators. These three different graded sets of integrals generate two different nonlinear, deformed forms of $su(2\mathrm{\mid }2)$, in which the Hamiltonian plays a role of a multiplicative central charge. On the ground state, the nonlinear superalgebra is reduced to the two distinct 2D Euclidean analogs of a superextended Poincar'e algebra used earlier in the literature for investigation of spontaneous supersymmetry breaking. We indicate that the observed exotic supersymmetric structure with three different ${\mathbb{Z}}_2$-gradings can be useful for the search of hidden symmetries in some other quantum systems, in particular, related to the Lam'e equation.