  In this paper we construct a family of commuting multidimensional
differential operators of order 3, which is closely related to the KdV
hierarchy. We find a common eigenfunction of this family and an algebraic
relation between these operators. Using these operators we associate a
hyperelliptic curve to any solution of the stationary KdV equation. A basic
generating function of the solutions of stationary KdV equation is introduced
as a special polarization of the equation of the hyperelliptic curve. We also
define and discuss the notion of a $w$-function of a solution of the stationary
$g$-KdV equation.
