  We exhibit a basis for the space of spherical characters of a distinguished
supercuspidal representation $\pi$ of a connected reductive $p$-adic group,
subject to the assumption that $\pi$ is obtained via induction from a
representation of an open compact mod centre subgroup. We derive an integral
formula for each spherical character belonging to the basis. This formula
involves integration of a particular kind of matrix coefficient of $\pi$. We
also obtain a similar formula for the function realizing the spherical
character. In addition, we determine, subject to some conditions, which of
these spherical characters vanish identically on an open neighbourhood of the
identity. We verify that the requisite conditions are always satisfied for
distinguished tame supercuspidal representations of groups that split over
tamely ramified extensions.
