Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems

In this paper, we study properties of the algebras of planar quasi-invariants. These algebras are Cohen–Macaulay and Gorenstein in codimension one. Using the technique of matrix problems, we classify all Cohen–Macaulay modules of rank one over them and determine their Picard groups. In terms of this classification, we describe the spectral modules of the planar rational Calogero–Moser systems. Finally, we elaborate the theory of the algebraic inverse scattering method, providing explicit computations of some ‘isospectral deformations’ of the planar rational Calogero–Moser system in the case of the split rational potential.