We explicitly construct polynomial vector fields Lk, k = 0, 1, 2, 3, 4, 6, on the complex linear space C6 with coordinates X = (x2, x3, x4) and Z = (z4, z5, z6). The fields Lk are linearly independent outside their discriminant variety Δ ⊂ C6 and are tangent to this variety. We describe a polynomial Lie algebra of the fields Lk and the structure of the polynomial ring C[X,Z] as a graded module with two generators x2 and z4 over this algebra. The fields L1 and L3 commute. Any polynomial P(X,Z) ∈ C[X,Z] determines a hyperelliptic function P(X,Z)(u1, u3) of genus 2, where u1 and u3 are the coordinates of trajectories of the fields L1 and L3. The function 2x2(u1, u3) is a two-zone solution of the Korteweg–de Vries hierarchy, and ∂z4(u1, u3)/∂u1 = ∂x2(u1, u3)/∂u3.
|Number of pages||25|
|Journal||Proceedings of the Steklov Institute of Mathematics|
|Publication status||Published - 1 Aug 2016|