[@Wal1; @Wal2; @Wal3; @Wal4], Waldspurger studied Shimura correspondence between half integral weight and integral weight modular forms in the framework of automorphic representations of the metaplectic group $\mathrm{Mp}_2$. One of the remarkable results in these papers is a detailed description of the automorphic discrete spectrum of $\mathrm{Mp}_2$. Theta correspondence plays a central role in this representation theoretic treatment. In the course of the study, especially in [@Wal2; @Wal4], Waldspurger found a close relation among special values of $L$-functions, period integrals and nonvanishing of theta correspondence. Our research focuses on nonvanishing of the period integrals with an eye on the work of Waldspurger [@Wal4] which was extended by Friedberg and Hoffstein [@FH]. We briefly review these results.

Let $E$ be a quadratic étale algebra over a number field $F$. Let $\eta$ denote the quadratic character of $\A_F^\times/F^\times$ associated with $E$ via the global class field theory if $E$ is a field and the trivial character otherwise. We write the ring of adèles of $F$ by $\A=\A_F$ and set $\A_E=\A_F\otimes_FE$. We fix an embedding $E\hookrightarrow\M_2(F)$ and regard $\operatorname{Res}_{E/F}\bb{G}_{m, E}$ as an $F$-subgroup of $\GL_2$. Here, $\operatorname{Res}_{E/F}$ denotes the restriction of scalars.

We say that $\pi$ is $E^\times$-distinguished if the integral $$\mP_E(\vp)=\int_{E^\times Z(\A)\bs\A_E^\times}\vp(h)\d 