Polynomial dynamical systems and Korteweg--de Vries equation

In this work we explicitly construct polynomial vector fields $\mathcal{L}_k,\;k=0,1,2,3,4,6$ on the complex linear space $\mathbb{C}^6$ with coordinates $X=(x_2,x_3,x_4)$ and $Z=(z_4,z_5,z_6)$. The fields $\mathcal{L}_k$ are linearly independent outside their discriminant variety $\Delta \subset \mathbb{C}^6$ and tangent to this variety. We describe a polynomial Lie algebra of the fields $\mathcal{L}_k$ and the structure of the polynomial ring $\mathbb{C}[X, Z]$ as a graded module with two generators $x_2$ and $z_4$ over this algebra. The fields $\mathcal{L}_1$ and $\mathcal{L}_3$ commute. Any polynomial $P(X,Z) \in \mathbb{C}[X, Z]$ determines a hyperelliptic function $P(X,Z)(u_1, u_3)$ of genus $2$, where $u_1$ and $u_3$ are coordinates of trajectories of the fields $\mathcal{L}_1$ and $\mathcal{L}_3$. The function $2 x_2(u_1, u_3)$ is a 2-zone solution of the KdV hierarchy and $\frac{\partial}{\partial u_1}z_4(u_1, u_3)=\frac{\partial}{\partial u_3}x_2(u_1, u_3)$.