2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94509We present here the iteration procedure for the determination of free energy $ε^{2}$-expansion using the theory of KdV - type equations. In our approach we use the conservation laws for KdV - type equations depending explicitly on times $t_{1}, t_{2}, ...$ to find the $ε^{2}$-expansion of $u(x,t_{1},t_{2},...)$ after the infinite number of shifts of $u(x,0,0,...) \equiv x$ along $t_{1}, t_{2}, ...$ in recurrent form. The formulas for the free energy expansion are just obtained then as a result of quite simple integration procedure applied to $u_{n}(x)$.18 pages, LatexExactly Solvable and Integrable SystemsHigh Energy Physics - Theorytext