An analytical approximate solution of the nonlinear differential equation in the field of analytic
The class of non - linear differential equation, the solution of which has, a movable singularities, considered in this work. This fact does not allow the use of the classical theorem on the existence and development of the new version updates the existence and uniqueness of the solution that has been done, by the authors. On the basis, of a new existence and uniqueness theorems built analytic approximate solution of a class of the normal form of nonlinear third-order differential equation with a polynomial structure of the fourth degree in the field of analytic. The technology of the proof of existence of uniqueness in the future can meet the challenges arising in the method of approximate solution of nonlinear differential equations with moving singular points.