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) $mu =mu _{0}+left( texttt{h} -1right) p,~texttt{h} >1$. 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.
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