Lets 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-dimentional integral manifold $L$ of the Cartan distribution is called a multivalued solution of equation E, if $L\subset E$

(see A.G.Kushner paper about classification of monge ampere 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 $?

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 $?

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-dimentional integral manifold $L$ of the Cartan distribution is called a multivalued solution of equation E, if $L\subset E$

(see A.G.Kushner paper about classification of monge ampere 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 $?

Lets 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-dimentional integral manifold $L$ of the Cartan distribution is called a multivalued solution of equation E, if $L\subset E$

(see A.G.Kushner paper about classification of monge ampere 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 $?