Recall that a doubly-stochastic matrix is a square matrix with non-negative elements such the sum of the elements in every row, as well as in every column, is $1$. The set of doubly-stochastic matrices is described by the Birkhoff – von Neumann theorem. For the matrices of prime order $p\ge 3$, this theorem can be interpreted to describe the set of all functions $w\colon\mathbb F_p^2\to\mathbb R^+$ such that for some pair of fixed directions in $\mathbb F_p^2$, every line $l$ in any of these directions gets its exact share of the total mass of $w$: $$ \sum_{x\in l} w(x) = \frac1p\,\sum_{x\in\mathbb F_p^2 } w(x). \tag{$*$} $$
Let's say that a line $l\in\mathbb F_p^2$ is even if it satisfies ($*$), and that a direction in $\mathbb F_p^2$ is even if all $p$ lines in this directions are even. Suppose that, instead of just two even directions (as in the Birkhoff – von Neumann theorem), there are $k\ge 3$ even directions. Intuitively, one can expect much more structure in this case.
For a prime $p$ and integer $3\le k\le p$, what is the set of all functions $w\colon\mathbb F_p^2\to\mathbb R^+$ possessing at least $k$ even directions?
In particular, isn't this set of functions the matching polytope of some graph?
