Commuting differential operators in two-dimension

A generalization to the multi-dimensional case of commutative rings of differential operators is considered. An algorithm for construction of commuting two-dimensional differential operators is formulated for a special kind of operators related to the simple one-dimensional model proposed by Burchnall and Chaundy in 1932. The problem of classifying such commutative pairs is discussed. The suggested algorithm is based on necessary conditions for general commutativity and the reducibility lemma proved in the present paper.