  We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical
systems on Riemann surfaces subjected to an external magnetic field. The
realization is shown to be possible only for Riemann surfaces with constant
curvature metrics. The cases of the sphere and Lobachevski plane are elaborated
in detail. The partial algebraization of the spectrum of the corresponding
Hamiltonians is proved by the reduction to one-dimensional quasi-exactly
solvable sl(2,R) families. It is found that these families possess the
"duality" transformations, which form a discrete group of symmetries of the
corresponding 1D potentials and partially relate the spectra of different 2D
systems. The algebraic structure of the systems on the sphere and hyperbolic
plane is explored in the context of the Onsager algebra associated with the
nonlinear holomorphic supersymmetry. Inspired by this analysis, a general
algebraic method for obtaining the covariant form of integrals of motion of the
quantum systems in external fields is proposed.
