mapping $\varphi\colon \mathcal{B}_g\times \mathbb{C}^g \to \mathcal{U}_g$, which defines the projection $\lambda\times \mathbb{C}^g \to \mathbb{C}^g/\Gamma_g(\lambda)$ for any $\lambda\in \mathcal{B}_g$. Let us fix the coordinates $(\lambda;u)$ in $\mathcal{B}_g\times \mathbb{C}^g\subset \mathbb{C}^{2g}\times \mathbb{C}^g$ where $u = (u_1,\ldots,u_{2g-1})$. Thus, using the mapping $\varphi$, we fixed in $\mathcal{U}_g$ the structure of the space of the bundle whose fibers $J_\lambda$ are principally polarized Abelian varieties.

We denote by $F = F_g$ the field of functions on $\mathcal{U}_g$ such that for any $f\in F$ the function $\varphi^*(f)$ is meromorphic, and its restriction to the fiber $J_\lambda$ is an Abelian function for any point $\lambda \in \mathcal{B}_g$.

Below, we will identify the field $F$ with its image in the field of meromorphic functions on $\mathcal{B}\times \mathbb{C}^g$.

The following [**Problem I**]{}:

[*Describe the Lie algebra of differentiations of the field of meromorphic functions on $\mathcal{B}_g\times \mathbb{C}^g$, generated by the operators $L$, such that $Lf\in F$ for any function $f\in F$*]{}

was solved in [@bl07; @bl08].

From the differential geometric point of view, Problem I is closely related to [**Problem II**]{}:

[*Describe the connection of the bundle $\pi\colon \mathcal{U}_g \to \mathcal{B}_g$.*]{}

The solution of Problem II leads to an important class of solutions of well-known equations of mathematical physics. In the 