and given them a lot of suggestions and advices.

The area under consideration is the construction of Abelian functions in terms of multi-variable $\sigma$-functions. Similarly to the Weierstrass elliptic function, the multi-variable sigma keeps the same main property - it remains form-invariant at the action of the symplectic group. Abelian functions appear as logarithmic derivatives like the $\zeta, \wp$-functions of the Weierstrass theory and similarly to the standard theta-functional approach which lead to the Krichever formula for KP solutions, [@kr1977]. But a fundamental difference between sigma and theta-functional theories is the following. They are both constructed by the curve and given as series in Jacobian variables, but in the first case the expansion is purely algebraic with respect to the model of the curve since its coefficients are polynomials in parameters of the defining curve equation, whilst in the second case coefficients are transcendental, being built in terms of Riemann matrix periods which are complete Abelian integrals.

In many publications, in particular see [@oni05; @eemop07; @bel12; @ny12; @ehkklp12; @an13; @bef13; @mp14; @nak16; @nak18] and references therein, it was demonstrated that multi-variable $\wp$-functions represent a language which is very suitable to speak about completely integrable systems of KP type. In particular, 