  We continue the work of [1, 2, 3] by analyzing the equivalence relation of
bi-embeddability on various classes of countable planes, most notably the class
of countable non-Desarguesian projective planes. We use constructions of the
second author to show that these equivalence relations are invariantly
universal, in the sense of [3], and thus in particular complete analytic. We
also introduce a new kind of Borel reducibility relation for standard Borel
G-spaces, which requires the preservation of stabilizers, and explain its
connection with the notion of full embeddings commonly considered in category
theory.
