On the existence of solutions of the system of Peterson–Codazzi and gauss equations

This paper is concerned with isometric embeddings of complete two-dimensional metrics, defined on the plane, whose curvature is bounded by negative constants (metrics of type L). It is proved that under certain conditions any horocycle in a metric of type L (an analog of a horocycle in the Lobachevskii plane) admits a $C^3$-isometric embedding into $E^3$. The proof is based on the construction of a smooth solution of the system of Peterson–Codazzi and Gauss equations in an infinite domain.