  A fundamental question in Celestial Mechanics is to analyze the possible
final motions of the Restricted $3$-body Problem, that is, to provide the
qualitative description of its complete (i.e. defined for all time) orbits as
time goes to infinity. According to the classification given by Chazy back in
1922, a remarkable possible behaviour is that of oscillatory motions, where the
motion $q$ of the massless body is unbounded but returns infinitely often
inside some bounded region: \[ \limsup_{t\to\pm\infty}
|q(t)|=\infty\qquad\qquad\text{and}\qquad\qquad \liminf_{t\to\pm\infty}
|q(t)|<\infty. \] In contrast with the other possible final motions in Chazy's
classification, oscillatory motions do not occur in the $2$-body Problem, while
they do for larger numbers of bodies. A further point of interest is their
appearance in connection with the existence of chaotic dynamics.
  In this paper we introduce new tools to study the existence of oscillatory
motions and prove that oscillatory motions exist in a particular configuration
known as the Restricted Isosceles $3$-body Problem (RI3BP) for almost all
values of the angular momentum. Our method, which is global and not limited to
nearly integrable settings, extends the previous results
\cite{guardia2021symbolic} by blending variational and geometric techniques
with tools from nonlinear analysis such as topological degree theory. To the
best of our knowledge, the present work constitutes the first complete analytic
proof of existence of oscillatory motions in a non perturbative regime.
