On regular embedding integrally in $R^3$ of metrics of class $C^4$ of negative curvature

On the $x_0y$ plane let there be specified a complete metric of negative curvature $K$ by means of the line element
$$ds^2=dx^2+B^2(x,y)\,dy^2$$
, and, in the strip $\Pi_a=\{0\le x\le a,-\infty<y<+\infty\}$, let the following conditions be met: $B(x,y)$ is a $C^4$-bounded function $B\ge\lambda>0$, $K\le-\mu^2<0$ ($\lambda$ and $\mu$ are constants). Then, the metric in strip $\Pi_a$ is embedded in $R^3$ by means of a surface of class C3.