elements $e_i$ $ i\in \{1, \ldots, 2g+1 \} -  \{ i_r,i_s  \}  $

Let us analyse equations (\[Kleineq\]) for small genera, $g\leq 5$. One can see that plugging to the equation (\[JIP1\]) to (\[Kleineq\]) we get non-homogeneous linear equations solvable by Kramer rule and the solutions can be presented in the form (\[formula\]).

Now suppose that (\[formula\]) is valid for higher $g>5$ were computer power is insufficient to check that by means of computer algebra. But that’s possible to check (\[formula\]) for arbitrary big genera numerically giving to branch points $e_i, i=1,\ldots, i=2g+1$ certain numeric values. Many checking confirmed (\[formula\]) .

Quantities $\wp_{i,j}({\Omega}_{\mathcal{I}_0})$ are expressed in terms of even $\theta$-constants as follows $$\begin{aligned}
 \wp_{i,j}({\Omega}_{\mathcal{I}_0})=-2\varkappa_{i,j}
- \frac{1}{\theta[\varepsilon_{\mathcal{I}_0}]({0})}
\partial_{{U}_i,{U_j} }^2 \theta[\varepsilon_{\mathcal{I}_0}]({0}), \quad \forall \mathcal{I}_0,\; i,j=1,\ldots, g.\end{aligned}$$ Here $[\varepsilon_{\mathcal{I}_0}]$ is characteristic of the vector $\left[{\Omega}_{\mathcal{I}_0}+{K}_{\infty}  \right]$, where ${K}_{\infty}$ is vector of Riemann constants with base point $\infty$ and $\partial_{{U}}$ is directional derivative along vector $\boldsymbol{U}_i$, that is $i$th column vector of inverse matrix of $\mathfrak{a}$-periods, $\mathcal{A}^{-1} = ( {U}_1,\ldots, {U}_g )$. The same formula is valid for all possible partitions $\mathcal{I}_0\cup \mathcal{J}_0$, there are $N_g$ of that, that is number of non-singular even characteristics, $$N_g= \left( \begin{array}{c}  2g+1\\ g  \end{array}  \right)$$ Therefore one 