case $g=1$, the solution is called the Frobenius-Stikelberger connection (see [@dubr96]) and leads to solutions of Chazy equation.

The space $\mathcal{U}_g$ is a rational variety, more precisely, there is a birational isomorphism $\varphi\colon \mathbb{C}^{3g} \to \mathcal{U}_g$. This fact was discovered by B. A. Dubrovin and S. P. Novikov in [@dn974]. In [@dn974], a fiber of the universal bundle is considered as a level surface of the integrals of motion of $g$th stationary flow of KdV system, that is, it is defined in $\mathbb{C}^{3g}$ by a system of $2g$ algebraic equations. The degree of the system grows with the growth of genus. In [@bel97], [@bel997] the coordinates in $\mathbb{C}^{3g}$ were introduced such that a fiber is defined by $2g$ equations of degree not greater than $3$.

The Dubrovin-Novikov coordinates and the coordinates from [@bel97], [@bel997] are the same for the universal space of genus $1$ curves. But already in the case of genus 2, these coordinates differ (see [@bl08]).

The integrals of motion of KdV systems are exactly the coefficients $\lambda_{2g+4},\ldots,\lambda_{4g+2}$ of hyperelliptic curve $V_\lambda$, in which the coefficients $\lambda_{4},\ldots,\lambda_{2g+2}$ are free parameters (see ). Choosing a point $z\in \mathbb{C}^{3g}$ such that the point $\varphi(z)\in \mathcal{B}_g$ is defined, one can calculate 