Vector rank of commuting matrix differential operators. Proof of S. P. Novikov's criterion

The problem of describing a commuting pair of differential operators in terms of its Burchnall–Chaundy curve and the holomorphic bundle over it is considered. A characteristic of the matrix case is the occurrence of vector rank, a bundle having various dimensions over various components of the Burchnall–Chaundy curve. A complete, independent system which determines the pair of operators uniquely is chosen. In the last section, a proof is given of S. P. Novikov's criterion for an operator with periodic coefficients to be an operator of a nontrivial commuting pair.
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