  We study the spatial isosceles three body problem, which is a system with two
degrees of freedom after modulo the rotation symmetry. For certain choices of
energy and angular momentum, we find some disk-like global surfaces of section
with the Euler orbit as their common boundary, and a brake orbit passing
through them. By considering the Poincar\'e maps of these global surfaces of
section, we prove the existence of all kinds of different periodic orbits under
certain assumption. Moreover, we are able to prove, for generic choices of
masses, the system always has infinitely many periodic orbits.
  One of the key is to estimate the rotation numbers of the Euler orbit and the
brake orbit with respect to the Poincar\'e map. For this, we establish formulas
connected these numbers with the mean indices of the corresponding orbits using
the Maslov-type index.
