  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
$\frac{\partial}{\partial u_1}z_4(u_1, u_3)=\frac{\partial}{\partial
u_3}x_2(u_1, u_3)$.
