Solving N for HN=0, Given H is a special type of skew symmetric (n x n, n is a odd number) matrix.

Question

Solving $N\ \mathrm{for}\ H \times N =0$, given $H$ is a special type of skew symmetric matrix $(n \times n, n\ \mathrm{is\ an\ odd\ number}\ n=2k+1)$, 0 on diagonal and 1, -1 in off-diagonal entries. Each row and column in H has exactly k 1s and k -1s and one 0. N is a $n \times 1$ vector satisfies $N' \times 1\ n=1$. All entries in N is non-negative. Naturally, N' = (1/n, 1/n, ..., 1/n) is one of the solutions, my question is there any other solutions or this solution is a unique one? If so, how to prove this?