On complete convex solutions of the equation $\operatorname{spur}_m(z_{ij})=1$

Let designation $\operatorname{spur}_m(z_{ij})=1$ stand for the sum of all principal $m$-order minors of matrix $(z_{ij})$, consisting of second derivatives of the function $z(x^1,\dots,x^n)$. Any complete convex class $C^{2\alpha}$ solution of the equation $\operatorname{spur}_m(z_{ij})=1$, ($2\le m<n$), will be a quadratic polynomial if the matrix $(z_{ij})$ eigenvalues are sufficiently close to each other.