  The singular real second order 1D Schrodinger operators are considered here
with such potentials that all local solutions near singularities to the
eigenvalue problem are meromorphic for all values of the spectral parameter.
All algebro-geometrical or "singular finite-gap" potentials satisfy to this
condition. A Spectral Theory is constructed here for the periodic and rapidly
decreasing cases in the special classes of functions with singularities and
indefinite inner product. It has a finite number of negative squares if the
unimodular Bloch multipliers are fixed in the periodic case, and in the rapidly
decreasing case. The time dynamics provided by the KdV hierarchy preserves this
number. The right analog of Fourier Transform for the Riemann Surfaces
preserving remarkable multiplicative properties of the ordinary (i.e. genus
zero) Fourier Transform based on the standard exponential basis, leads to such
operators as it was shown in our previous works.
