  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.
