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

Comment: 8 pages

• 14D21 - Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
• 37-XX - Dynamical systems and ergodic theory
• 37K10 - Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
• 37K20 - Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
• 81-XX - Quantum theory