  Full manifold of the complex Bloch-Floquet eigenfunctions is investigated for
the ground level of the purely magnetic 2D Pauli operators (equal to zero
because of supersymmetry). Deep connection of it with the 2D analog of the
"Burgers Nonlinear Hierarchy" plays fundamental role here. Everything is
completely calculated for the broad class of Algebro-Geometric operators found
in this work for this case. For the case of nonzero flux the ground states were
found by Aharonov-Casher (1979) for the rapidly decreasing fields, and by
Dubrovin-Novikov (1980) for the periodic fields. No Algebro-Geometric operators
where known in the case of nonzero flux. For genus $g=1$ we found periodic
operators with zero flux, singular magnetic fields and Bohm-Aharonov
phenomenon. Our arguments imply that the delta-term really does not affect
seriously the spectrum nearby of the ground state. For $g>1$ our theory
requires to use only algebraic curves with selected point leading to the
solutions elliptic in the variable $x$ for KdV and KP in order to get periodic
magnetic fields. The algebro-geometric case of genus zero leads, in particular,
to the slowly decreasing lump-like magnetic fields with especially interesting
variety of ground states in the Hilbert Space $\cL_2(\bR^2)$.
