of coordinates $P,Q$ and necessarily symmetric matrix $\varkappa^T=\varkappa$, $\varkappa=\eta(2\omega)^{-1}$ is introduced to satisfy the normalization condition ${\bf ii}$. In shorter form (\[omega1\]) cab be rewritten as $$\Omega(P,Q) =\frac{2 yw +F(x,z)}{ 4 (x-z)^2  yw } \mathrm{d}x\mathrm{d}z + 2\mathrm{d}{u}^T (P)\varkappa
\mathrm{d}{u} (Q)\label{omega1}$$ where $F(x,z)$ is so-called Kleinian 2-polar, given as $$F(x,z)=\sum_{k=0}^g x^kz^k  \left( 2\lambda_{2k}+\lambda_{2k+1}(x+z)  \right) \label{polar}$$ Recently algebraic representation for $\Omega(P,Q)$ similar to (\[omega1\]) found in [@suz17], [@eyn18] for wide class on algebraic curves, included $(n,s)$-curves [@bel999].

Main relation lying in the base of the theory is Riemann formula represented third kind Abelian integral as $\theta$-quotient written in terms of described above realization of the fundamental differential $\Omega(P,Q)$.

(Riemann) Let $P'=(x',y')$ and $P''=(x'',y'')$ are two arbitrary distinct points on $\mathcal{C}$ and let $\mathcal{D}'=\{ P_1'+\ldots+P_{g}'\}$ and $\mathcal{D}''=\{ P_1''+\ldots+P_{g}''\}$ are two non-special divisors of degree $g$. Then the following relation is valid $$\begin{aligned}
\begin{split}
&\int_{P''}^{P'} \sum_{j=1}^g\int_{P_j'}^{P_j''} \left\{  \frac{2yy_i+F(x,x_i)}{4(x-x_i)^2}\frac{\mathrm{d}x}{y}\frac{\mathrm{d}x_i}{y_i} + 2 \mathrm{d}{u}(x,y)\varkappa \mathrm{d} {u}(x_i,y_i)   \right\}  \\
&=\mathrm{ln}
\left(\frac{  \theta(\mathcal{A}(P')-\mathcal{A}(\mathcal{D}') +{K}_{\infty} )}
           { \theta(\mathcal{A}(P')-\mathcal{A}(\mathcal{D}'') +{K}_{\infty} ) }       \right)
-\mathrm{ln}
\left(\frac{\theta(\mathcal{A}(P'')-\mathcal{A}(\mathcal{D}') +{K}_{\infty} )}
           {\theta(\mathcal{A}(P'')-\mathcal{A}(\mathcal{D}'') +{K}_{\infty} ) } \right)
\end{split} \label{Riemann1}\end{aligned}$$ where $\mathcal{A}(P)= \int_{\infty}^{P} \mathrm{d}{v}$ Abel map with base point $\infty$, ${K}_{\infty}$ - vector of Riemann constants with bases point $\infty$ which is a half-period.

Introduce multi-variable fundamental 