  Let $E/F$ be a quadratic extension of number fields and let $\pi$ be an
$\mathrm{SL}_n(\mathbb{A}_F)$-distinguished cuspidal automorphic representation
of $\mathrm{SL}_n(\mathbb{A}_E)$. Using an unfolding argument, we prove that an
element of the $\mathrm{L}$-packet of $\pi$ is distinguished if and only if it
is $\psi$-generic for a non-degenerate character $\psi$ of $N_n(\mathbb{A}_E)$
trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper
triangular matrices of $\mathrm{SL}_n$. We then use this result to analyze the
non-vanishing of the period integral on different realizations of a
distinguished cuspidal automorphic representation of
$\mathrm{SL}_n(\mathbb{A}_E)$ with multiplicity $> 1$, and show that in general
some canonical copies of a distinguished representation inside different
$\mathrm{L}$-packets can have vanishing period. We also construct examples of
everywhere locally distinguished representations of
$\mathrm{SL}_n(\mathbb{A}_E)$ the $\mathrm{L}$-packets of which do not contain
any distinguished representation.
