  When can a unimodular random planar graph be drawn in the Euclidean or the
hyperbolic plane in a way that the distribution of the random drawing is
isometry-invariant? This question was answered for one-ended unimodular graphs
in \cite{benjamini2019invariant}, using the fact that such graphs automatically
have locally finite (simply connected) drawings into the plane. For the case of
graphs with multiple ends the question was left open. We revisit Halin's graph
theoretic characterization of graphs that have a locally finite embedding into
the plane. Then we prove that such unimodular random graphs do have a locally
finite invariant embedding into the Euclidean or the hyperbolic plane,
depending on whether the graph is amenable or not.
