  We show that the N=2 superextended 1D quantum Dirac delta potential problem
is characterized by the hidden nonlinear $su(2|2)$ 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 $su(2|2)$, 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.
