$$\begin{aligned}
\Omega_{\mathcal{I}_0} = \int_{\infty}^{( e_{i_1},0  )}\mathrm{d} {u}
+\ldots+  \int_{\infty}^{( e_{i_g},0  )}\mathrm{d} {u},\quad  \mathcal{I}_0= \{i_1,\ldots, i_g\}\subset \{1,\ldots,2g+1\}
 \end{aligned}$$

Denote elementary symmetric functions $s_n(\mathcal{I}_0)$, $S_{n}(\mathcal{J}_0)$ of order $n$ built on branch points $\{ e_{i_k} \}$, $i_k\in \mathcal{I}_0$, $\{ e_{j_k} \}$, $j_k\in \mathcal{J}_0$ correspondingly. In particular, $$\begin{aligned}
\begin{split}
s_1(\mathcal{I}_0)&=e_{i_1}+\ldots+ e_{i_g}, \hskip1.65cm S_1(\mathcal{J}_0)=e_{j_1}+\ldots+ e_{j_{g+1}}\\
s_2(\mathcal{I}_0)&=e_{i_1}e_{i_2}+\ldots+e_{i_{g-1}} e_{i_g},
\hskip0.5cm S_2(\mathcal{J}_0)=e_{j_1}e_{j_2}+\ldots+e_{j_{g}} e_{i_{g+1}}\\
&\vdots \hskip5.4cm\vdots\\
s_g(\mathcal{I}_0)&=e_{i_1}\cdots e_{i_g} \hskip 2.62cm
S_{g+1}(\mathcal{J}_0)=e_{j_1}\cdots e_{j_{g+1}}
\end{split}\end{aligned}$$ Because of symmetry, $\wp_{p,q}({\Omega}_{\mathcal{I}_0})=\wp_{q,p}({\Omega}_{\mathcal{I}_0})$ enough to find these quantities for $ p\leq q \in \{1,\ldots,g\}  $. The following is valid

(Conjectural Proposition ) Let even non-singular half-period ${\Omega}_{\mathcal{I}_0}$ associate to a partition $\mathcal{I}_0\cup \mathcal{J}_0=\{1,\ldots,2g+1\}$. Then for all $k,j\geq k, k,j = 1\ldots,g$ the following formula is valid

$$\begin{aligned}
\begin{split}
&\wp_{k,j}({\Omega}_{\mathcal{I}_0})\\&=(-1)^{k+j}\sum_{n=1}^k n \left( s_{g-k+n}(\mathcal{I}_0)S_{g-j-n+1}(\mathcal{J}_0)
+s_{g-j-n}(\mathcal{I}_0)S_{g+n-k+1}(\mathcal{J}_0) \right),
 \end{split} \label{formula}\end{aligned}$$

Klein formula written for even non-singular half period ${\Omega}_{\mathcal{I}_0}$ leads to linear system of equations with respect to Kleinian two-index symbols $ \wp_{i,j}({\Omega}_{\mathcal{I}_0}) $ $$\begin{aligned}
\sum_{i=1}^g\sum_{j=1}^g \wp_{i,j}({\Omega}_{\mathcal{I}_0})   e_{i_r}^{i-1}e_{i_s}^{j-1} =\frac{F(e_{i_r}, e_{i_s})}
{4(e_{i_r} - e_{i_s})^2}\quad  \; i_r, i_s \in \mathcal{I}_0\label{Kleineq}\end{aligned}$$

To solve these equation we note that $$\wp_{k,g}({\Omega}_{\mathcal{I}_0})= (-1)^{k+1}s_k(\mathcal{I}_0), \quad k=1,\ldots,g$$ Also note that $F(e_{i_r}, e_{i_s})$ is divisible by $(e_{i_r}- e_{i_s})^2$ and $$\frac{F(e_{i_r}, e_{i_s})}{4(e_{i_r} - e_{i_s})^2}= e_{i_r}^{g-1} e_{i_s}^{g-1}\mathfrak{S}_1+
e_{i_r}^{g-2} e_{i_s}^{g-2}\mathfrak{S}_2+\ldots + \mathfrak{S}_{2g-1}\label{JIP1}$$ where $\mathfrak{G}_k$ are order $k$ elementary symmetric functions of 