Implicit Function Theorem in a Neighborhood of an Abnormal Point

We study the existence of an implicit function, defined by an equation $G(x,\sigma )=0$, in a neighborhood of an abnormal point $(x_0,\sigma _0)$. We prove that if some $\lambda $-truncation of the mapping $F(x) = G(x,\sigma _0)$ is regular in a certain direction, then the sought implicit function exists.