  We investigate the existence of analytic solutions for the field equations in
the Einstein-\ae ther theory for a static spherically symmetric spacetime and
provide a detailed dynamical system analysis of the field equations. In
particular, we investigate if the gravitational field equations in the
Einstein-\ae ther model in the static spherically symmetric spacetime possesses
the Painlev\`e property, so that an analytic explicit integration can be
performed. We find that analytic solutions can be presented in terms of Laurent
expansion only when the matter source consists of a perfect fluid with linear
equation of state (EoS) $\mu =\mu _{0}+\left( \texttt{h} -1\right)
p,~\texttt{h} >1$. In order to study the field equations we apply the
Tolman-Oppenheimer-Volkoff (TOV) approach and other approaches. We find that
the relativistic TOV equations are drastically modified in Einstein-\ae ther
theory, and we explore the physical implications of this modification. We study
perfect fluid models with a scalar field with an exponential potential. We
discuss all of the equilibrium points and discuss their physical properties.
