## Abstract

We explicitly construct polynomial vector fields L_{k}, k = 0, 1, 2, 3, 4, 6, on the complex linear space C^{6} with coordinates X = (x_{2}, x_{3}, x_{4}) and Z = (z_{4}, z_{5}, z_{6}). The fields L_{k} are linearly independent outside their discriminant variety Δ ⊂ C^{6} and are tangent to this variety. We describe a polynomial Lie algebra of the fields L_{k} and the structure of the polynomial ring C[X,Z] as a graded module with two generators x_{2} and z_{4} over this algebra. The fields L_{1} and L_{3} commute. Any polynomial P(X,Z) ∈ C[X,Z] determines a hyperelliptic function P(X,Z)(u_{1}, u_{3}) of genus 2, where u_{1} and u_{3} are the coordinates of trajectories of the fields L_{1} and L_{3}. The function 2x_{2}(u_{1}, u_{3}) is a two-zone solution of the Korteweg–de Vries hierarchy, and ∂z_{4}(u_{1}, u_{3})/∂u_{1} = ∂x_{2}(u_{1}, u_{3})/∂u_{3}.

Original language | English |
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Pages (from-to) | 176-200 |

Number of pages | 25 |

Journal | Proceedings of the Steklov Institute of Mathematics |

Volume | 294 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Aug 2016 |

Externally published | Yes |