The nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem. Any classical supersymmetric system characterized by the nonlinear in the Hamiltonian superalgebra is symplectomorphic to a supersymmetric canonical system with the holomorphic form of the supercharges. Depending on the behaviour of the superpotential, the canonical supersymmetric systems are separated into the three classes. In one of them the parameter specifying the supersymmetry order is subject to some sort of classical quantization, whereas the supersymmetry of another extreme class has a rather fictive nature since its fermion degrees of freedom are decoupled completely by a canonical transformation. The nonlinear supersymmetry with polynomial in momentum supercharges is analysed, and the most general one-parametric Calogero-like solution with the second order supercharges is found. Quantization of the systems of the canonical form reveals the two anomalyfree classes, one of which gives rise naturally to the quasi-exactly solvable systems. The quantum anomaly problem for the Calogero-like models is "cured" by the specific superpotential-dependent term of order 2 . The nonlinear supersymmetry admits the generalization to the case of two-dimensional systems. *

The nonlinear n-supersymmetry with holomorphic supercharges is investigated for the 2D system describing the motion of a charged spin-1/2 particle in an external magnetic field. The universal algebraic structure underlying the holomorphic nsupersymmetry is found. It is shown that the essential difference of the 2D realization of the holomorphic n-supersymmetry from the 1D case recently analysed by us consists in appearance of the central charge entering non-trivially into the superalgebra. The relation of the 2D holomorphic n-supersymmetry to the 1D quasi-exactly solvable (QES) problems is demonstrated by means of the reduction of the systems with hyperbolic or trigonometric form of the magnetic field. The reduction of the n-supersymmetric system with the polynomial magnetic field results in the family of the one-dimensional QES systems with the sextic potential. Unlike the original 2D holomorphic supersymmetry, the reduced 1D supersymmetry associated with x 6 + ... family is characterized by the non-holomorphic supercharges of the special form found by Aoyama et al. *

Recently, it was noticed by us that the nonlinear holomorphic supersymmetry of order n ∈ N, n > 1, (n-HSUSY) has an algebraic origin. We show that the Onsager algebra underlies n-HSUSY and investigate the structure of the former in the context of the latter. A new infinite set of mutually commuting charges is found which, unlike those from the Dolan-Grady set, include the terms quadratic in the Onsager algebra generators. This allows us to find the general form of the superalgebra of n-HSUSY and fix it explicitly for the cases of n = 2, 3, 4, 5, 6. The similar results are obtained for a new, contracted form of the Onsager algebra generated via the contracted Dolan-Grady relations. As an application, the algebraic structure of the known 1D and 2D systems with n-HSUSY is clarified and a generalization of the construction to the case of nonlinear pseudo-supersymmetry is proposed. Such a generalization is discussed in application to some integrable spin models and with its help we obtain a family of quasi-exactly solvable systems appearing in the P T -symmetric quantum mechanics. *