We give a natural generalization of the classification of commutative rings of ordinary differential operators, given in works of Krichever, Mumford, Mulase, and determine commutative rings of operators in a completed ring of partial differential operators in two variables (satisfying certain mild conditions) in terms of Parshin's generalized geometric data. It uses a generalization of M.Sato's theory and is constructible in both ways.

Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of partial differential operators, and we investigate the properties of these geometric data. This construction is in some sense similar to the construction of a formal module of Baker-Akhieser functions. On the other hand, there is a recent generalization of Sato's theory which belongs to the third author of this paper. We compare both approaches to the commutative rings of partial differential operators in two variables. As a by-product we get several necessary conditions on geometric data describing commutative rings of partial differential operators.

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In this paper we study rank two commuting ordinary differential operators with polynomial coefficients and the orbit space of the automorphisms group of the first Weyl algebra on such operators. We prove that for arbitrary fixed spectral curve of genus one the space of orbits is infinite. Moreover, we prove in this case that for for any n ≥ 1 there is a pair of selfadjoint commuting ordinary differential operators of rank two L 4 = (∂ 2x + V (x)) 2 + W (x) , L 6 , where W (x), V (x) are polynomials of degree n and n + 2 . We also prove that there are hyperelliptic spectral curves with the infinite spaces of orbits.