  We investigate all static spherically symmetric solutions in the context of
general relativity surrounded by a minimally-coupled quintessence field, using
dynamical system analysis. Applying the 1+1+2 formalism and introducing
suitable normalized variables involving the Gaussian curvature, we were able to
reformulate the field equations as first order differential equations. In the
case of a massless canonical scalar field we recovered all known black hole
results, such as the Fisher solution, and we found that apart from the
Schwarzschild solution all other solutions are naked singularities.
Additionally, we identified the symmetric phase space which corresponds to the
white hole part of the solution and in the case of a phantom field, we were
able to extract the conditions for the existence of wormholes and define all
possible class of solutions such as Cold Black holes, singular spacetimes and
wormholes like Ellis wormhole, for example. For an exponential potential, we
found that the black hole solution which is asymptotically flat is unique and
it is the Schwarzschild spacetime, while all other solutions are naked
singularities. Furthermore, we found solutions connecting to a white hole
through a maximum radius, and not a minimum radius (throat) such as wormhole
solutions, therefore violating the flare-out condition. Finally, we have found
a necessary and sufficient condition on the form of the potential to have an
asymptotically AdS spacetime along with a necessary condition for the existence
of asymptotically flat black holes.
