A Modified Regularization Method

We consider the Cauchy problem for the systems of the first-order ordinary differential equations with a small parameter E of degree q at the derivative. We study the possibility of solving this problem by means of the regularization method of the Lomov theory of singular perturbations. We show that, for q>1, an application of the procedure by Lomov leads only to the trivial solution to the problem in the class of resonance-free solutions. We suggest and describe a modification of the procedure which enables us to construct a nontrivial solution to the problem in the space of resonance-free solutions.