  Let E/F be a quadratic extension of number fields. For a cuspidal
representation $\pi$ of SL(2,A_E), we study the non-vanishing of the period
integral on SL(2,F)\SL(2,A_F). We characterise the non-vanishing of the period
integral of $\pi$ in terms of $\pi$ being generic with respect to characters of
E\A_E which are trivial on A_F. We show that the period integral in general is
not a product of local invariant functionals, and find a necessary and
sufficient condition when it is. We exhibit cuspidal representations of
SL(2,A_E) whose period integral vanishes identically while each local
constituent admits an SL(2)-invariant linear functional. Finally, we construct
an automorphic representation $\pi$ on SL(2,A_E) which is abstractly SL(2,A_F)
distinguished but none of the elements in the global L-packet determined by
$\pi$ is distinguished by SL(2,A_F).
