  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, computing a new unexpected explicit
example of a deformed Calogero-Moser system.
