Geodesic
Vector rank of commuting matrix differential operators. Proof of S.\,P.~Novikov's criterion
Izvestiya. Mathematics , Tome 28 (1987) no. 3, pp. 445-465

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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. Bibliography: 25 titles. 
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     author = {P. G. Grinevich},
     title = {Vector rank of commuting matrix differential operators. {Proof} of {S.\,P.~Novikov's} criterion},
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P. G. Grinevich. Vector rank of commuting matrix differential operators. Proof of S.\,P.~Novikov's criterion. Izvestiya. Mathematics , Tome 28 (1987) no. 3, pp. 445-465. http://geodesic.mathdoc.fr/item/IM2_1987_28_3_a1/