## ] o c n m p [## Open Communications in Nonlinear Mathematical Physics |

Articles and Letters published during 2024 (not complete)

The generating series for the instanton contribution to Green functions ofthe $2D$ sigma model was found in the works of Schwarz, Fateev and Frolov. Weshow that this series can be written as a formal tau function of the two-sidedtwo-component KP hierarchy. We call it formal singular tau function becausethis tau function is a sum where each term is the infrared and ultravioletdivergent one exactly as the series found by the mentioned authors. However onecan regularize this singluar tau function and to obtain regular observables.This is because observables contains ratious of mentioned divergentexpressions. Thus, we enladge the families of tau functions to work with.

We obtain conditions, which when fulfilled, permit to transform the coordinates of a dynamical system into pairs of canonical ones for some Hamiltonian system. These conditions, restricted to the class of coordinate transformations which act on each coordinate independently, are greatly simplified. However, they are surprisingly successful in defining canonical coordinates and an associated Hamiltonian for several test examples. So, a method is proposed to exploit these simple transformations in a systematic manner.