  We study the linear periods on $GL_{2n}$ twisted by a character using a new
relative trace formula. We establish the relative fundamental lemma and the
transfer of orbital integrals. Together with the spectral isolation technique
of Beuzart-Plessis--Liu--Zhang--Zhu, we are able to compare the elliptic part
of the relative trace formulae and to obtain new results generalizing
Waldspurger's theorem in the $n=1$ case.
