Weakly connected systems of Monge–Amper elliptic equations and the problem of existence of $2$-surface in $E^{k+2}$ with given Killing–Lipschitz curvatures with respect to $k$ normal vectors

A surface $z^i=u^i(x,y)$, $i=1,\dots,k$, projected regularly onto a domain $\Omega$ of the $(x,y)$-plane is considered in a $(k+2)$-dimensional Euclidean space. We introduce natural unit vectors $\xi_i$ directed along the vectors $(u^i_x,u^i_y,0,\dots,0,-1,0,\dots)$, $i=1,\dots,k$, where $-1$ is in the $(2+i)$-coordinate place, and the Killing–Lipschitz curvatures $K^i (x, y)$ with respect to these normal vectors. The problem of construction of a surface with given positive functions $K^i(x,y)$ and a given boundary value $u^i|_{\partial\Omega}=\varphi^i(\sigma)$, where $\sigma$ is the parameter in the curve $\partial\Omega$, is solved.