Polynomial dynamical systems and the Korteweg–de Vries equation

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 are 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 the coordinates of trajectories of the fields $\mathcal L_1$ and $\mathcal L_3$. The function $2x_2(u_1,u_3)$ is a two-zone solution of the Korteweg–de Vries hierarchy, and $\partial z_4(u_1,u_3)/\partial u_1=\partial x_2(u_1,u_3)/\partial u_3$.