In this paper, the author constructs explicitly polynomial vector fields \(L_k\), \(k=0, 1, 2, 3, 4, 6\) 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 \(L_k\) are linearly independent outside their discriminant variety \(\Delta\subset \mathbb{C}^6\) and tangent to this variety. He describes a polynomial Lie algebra of the fields \(L_k\) 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 \(L_1\) and \(L_3\) commute. Any polynomial \(P(X, Z)\in\mathbb{C}[X, Z]\) determines a hyperelliptic function \(P(X, Z)(u1, u3)\) of genus \(2\), where u1 and \(u_3\) are coordinates of trajectories of the fields \(L_1\) and \(L_3\). The function \(2x_2(u_1, u_3)\) is a \(2\)-zone solution of the Korteweg-de Vries (KdV) hierarchy and \(\frac{\partial}{\partial u_1} z_4(u_1, u_3)=\frac{\partial}{\partial u_3} x_2(u_1, u_3)\). In general, the present work belongs to a large field of research at the junction of the analytic theory of Riemann surfaces, abelian functions, and the theory of integrable systems. The paper is supported by two appendices describing the necessary information about elliptic sigma functions and sigma-functions of genus two curves.