On the problem of the normal image of a complete surface of negative curvature

In the first part of this note an example is constructed of a surface $z=z(x,y)$ satisfying the equation
\begin{equation} z_{xx}z_{yy}-z^2_{xy}=-1, \end{equation}
in the entire $(x,y)$ plane, whose normal image is a half-plane.
In the second part it is shown that the class of all integral surfaces of equation (1) defined in the entire $x,y$-plane has the property that their normal images cannot be infinite strips between parallel lines. Hence it follows, using results of N. V. Efimov, that the normal images of the above surfaces may only be planes or half-planes.
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