SCHWEITZER 2009 Problem 10

Let $U\subset\mathbb R^n$ be an open set, and let $L: U\times\mathbb R^n\to\mathbb R$ be a continuous, in its second variable first order positive homogeneous, positive over $U\times (\mathbb R^n\setminus\{0\})$ and of $C^2$ -class Langrange function, such that for all $p\in U$ the Gauss-curvature of the hyper surface $$\{ v\in\mathbb R^n \mid L(p,v) = 1 \}$$ is nowhere zero. Determine the extremals of $L$ if it satisfies the following system $$\sum_{k = 1}^n y^k\partial_k\partial_{n + i}L = \sum_{k = 1}^n y^k\partial_i\partial_{n + k} L (i\in\{1,\dots,n\})$$ of partial differetial equations, where $y^k(u,v) : = v^k$ for $(u,v)\in U\times\mathbb R^k$ , $v = (v^1,\dots,v^k)$ .