  In this paper we consider finite-temperature holographic RG flows in $D=3$
$\mathcal{N}=(2,0)$ gauged truncated supergravity coupled to a sigma model with
a hyperbolic target space. In the context of the holographic duality, fixed
points (CFTs) at finite temperature are described by AdS black holes. We come
from the gravity EOM to a 3d autonomous dynamical system, which critical points
can be related to fixed points of dual field theories. Near-horizon black hole
solutions correspond to infinite points of this system. We use Poincar\'e
transformations to project the system on $\mathbf{R}^3$ into the 3d unit
cylinder such that the infinite points are mapped onto the boundary of the
cylinder. We explore numerically the space of solutions. We show that the exact
RG flow at zero temperature is the separatrix for asymptotically AdS black hole
solutions if the potential has one extremum, while for the potential with three
extrema the separatrices are RG flows between AdS fixed points. We find
near-horizon analytical solutions for asymptotically AdS black holes using the
dynamical equations. We also present a method for constructing full analytical
solutions.
