On the problem of first correction in soliton perturbation theory

For an integro-differential equation with rapidly oscillating kernel of Cauchy type it is proved that a solution exists and is uniformly bounded in the Holder norm with respect to a small parameter $\varepsilon$. This equation describes the essential part of the first correction in soliton perturbation theory. An asymptotic estimate is obtained for this correction, which implies that the soliton structure of the solution of an equation close to the integral equation is preserved for times $\sim\varepsilon^{-1}$.