  We present the construction of stationary boson-fermion spherically symmetric
configurations governed by Newtonian gravity. Bosons are described in the
Gross-Pitaevskii regime and fermions are assumed to obey Euler equations for an
inviscid fluid with polytropic equation of state. The two components are
coupled through the gravitational potential. The families of solutions are
parametrized by the central value of the wave function describing the bosons
and the central denisty of the fluid. We explore the stability of the solutions
using numerical evolutions that solve the time dependent
Schr\"odinger-Euler-Poisson system, using the truncation error of the numerical
methods as the perturbation. We find that all configurations are stable as long
as the polytropic equation of state (EoS) is enforced during the evolution.
When the configurations are evolved using the ideal gas EoS they all are
unstable that decay into a sort of twin solutions that approach a nearly
stationary configuration. We expect these solutions and their evolution serve
to test numerical codes that are currently being used in the study of Fuzzy
Dark Matter plus baryons.
