Let's start with a definition:

Definition: A scalar k-th order differential equation on a smooth manifold $M$,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $

for $\left | \sigma \right |\leqslant k $.

This equation can be viewed as hypersurface $E=F(x,u,p_\sigma )=0\subset J^kM$ in the space of k-jets.

An n-dimensional integral manifold $L$ of the Cartan distribution is called a **multivalued solution** of equation E, if $L\subset E$.

(see A.G. Kushner's paper about the classification of Monge-Ampère equations)

How can we find the multivalued solution of the equation $\frac{v_{xx}v_{yy}-v_{xy}^{2}}{(1+v_{x}^{2}+v_{y}^{2})^2}=1 $?

notcontact-equivalent to Laplace's equation (though it is, of course, elliptic); it is not even Darboux-integrable (again, Lie). Rather, it is contact equivalent to the elliptic sinh-Gordon equation $\Delta u = \mathrm{sinh}\ u$, which is an integrable equation. Of course, there are many multi-valued solutions, such as $v(x,y)=\pm\sqrt{1-x^2-y^2}$, and many can be found using integrable systems techniques. – Robert Bryant Mar 5 '13 at 13:21allof the multi-valued solutions explicitly in terms of two arbitrary functions of one variable (or one holomorphic function of one variable), since the equation is not Darboux-integrable. – Robert Bryant Mar 11 '13 at 12:47