On the regularity of oricyclic coordinates

Suppose there is defined in the plane a complete metric $W^-$, whose curvature $K$ satisfies the inequality $-k_2^2\le K\le -k_1^2$ ($k_1$ and $k_2$ are positive constants) and some regularity hypothesis. Then in the entire domain of definition of the metric $W^-$ one can construct regular oricyclic coordinates $(x,y)$, in which the line element has the form $ds^2=dx^2+B2(x,y)\cdot dy^2$.