  Let $F$ be a non-archimedean local field of characteristic zero. We study the
linear period problem for the pair $(G,H_{p,p+1})=(GL_{2p+1}(F),
GL_{p}(F)\times GL_{p+1}(F))$ and we prove that any
bi-$(H_{p,p+1},\mu)$-invariant generalized function on $G$ is invariant under
the matrix transpose when \mu is a good character. We also show that any $P\cap
H_{p,p+1}$-invariant linear functional on an $H_{p,p+1}$-distinguished
irreducible smooth representation of $G$ is also $H_{p,p+1}$-invariant when F
is nonarchimedean, where $P$ is a standard mirabolic subgroup of $G$ with last
row vector $(0,\cdots,0,1)$.
