The existence of real solutions of a nonlinear equation in a neighborhood of an abnormal (degenerate) point is studied. We prove that if the mapping describing this equation admits a lambda-truncation that is regular in some direction, then there exist solutions for all right-hand sides in some neighborhood of the image of the abnormal point. Under the same assumption, we prove that there exist nontrivial smooth curves passing through the abnormal point and lying on the level set of the map in question.