Nonlinear symmetries of perfectly invisible $PT$-regularized conformal and superconformal mechanics systems

We investigate how the Lax-Novikov integral in the perfectly invisible $PT$-regularized zero-gap quantum conformal and superconformal mechanics systems affects on their (super)-conformal symmetries. We show that the expansion of the conformal symmetry with this integral results in a nonlinearly extended generalized Shr"odinger algebra. The $PT$-regularized superconformal mechanics systems in the phase of the unbroken exotic nonlinear $\mathcal{N}=4$ super-Poincar'e symmetry are described by nonlinearly super-extended Schr"odinger algebra with the $osp(2\mathrm{\mid }2)$ sub-superalgebra. In the partially broken phase, the scaling dimension of all odd integrals is indefinite, and the $osp(2\mathrm{\mid }2)$ is not contained as a sub-superalgebra.