  In this paper we study branching laws for certain unitary representations.
This is done on the smooth vectors via a version of the {\it period integrals},
studied in number theory, and also closely connected to the {\it
symmetry-breaking operators}, introduced by T.~Kobayashi.
  We exhibit non-vanishing symmetry breaking operators for the restriction of a
representation $\Pi$ in the discrete spectrum for real hyperboloids to
representations of smaller orthogonal groups. In the last part we discuss some
conjectures for the restriction of representations in Arthur packets containing
the representation $\Pi$ and the corresponding Arthur-Vogan packets to smaller
orthogonal groups; these are inspired by the Gross-Prasad conjectures.
