### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 032, 15 pages      arXiv:1811.07138      https://doi.org/10.3842/SIGMA.2019.032

### Construction of Two Parametric Deformation of KdV-Hierarchy and Solution in Terms of Meromorphic Functions on the Sigma Divisor of a Hyperelliptic Curve of Genus 3

Takanori Ayano a and Victor M. Buchstaber b
a) Osaka City University, Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
b) Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Street, Moscow, 119991, Russia

Received November 21, 2018, in final form April 11, 2019; Published online April 27, 2019

Abstract
Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we constructed the field of meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3 and solutions of the systems for $g=3$ by these functions. In this paper, as an application of our previous results, we construct two parametric deformation of the KdV-hierarchy. This new system is integrated in the meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3. In Section 8 of our previous paper [Funct. Anal. Appl. 51 (2017), 162-176], there are miscalculations. In appendix of this paper, we correct the errors.

Key words: Abelian functions; hyperelliptic sigma functions; polynomial dynamical systems; commuting vector fields; KdV-hierarchy.

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