On a Commutative Ring of Two Variable Differential Operators with Matrix Coefficients

In this work, we construct commutative rings of two variable matrix differential operators that are isomorphic to a ring of meromorphic functions on a rational manifold obtained from the CP 1×CP 1 by identification of two lines with the pole on a certain rational curve. The commutation condition for differential operators is equivalent to a system of non-linear equations in the operators’ coefficients. For selected operator coefficients, the commutation equations reduce to known soliton equations such as the Korteweg-de Vries equation, the Kadomtsev-Petviashvili equation, the sin-Gordon equation and others. The problem of classifying commuting ordinary differential operators was solved in [1]. If two differential operators L1 = ∂ n x + un−1∂ n−1 x + . . .+ u0(x), L2 = ∂ m x + vm−1∂ m−1 x + . . .+ v0(x) commute, then by the Burchnall-Chaundy lemma [2] there exists a non-zero polynomial Q(λ, μ) of two commuting variables λ and μ such that