  Complex reflection groups of rank two are precisely the finite groups in the
family of groups that we call virtual complex reflection groups of rank two.
These groups are particular cases of $J$-groups as defined by Achar \& Aubert
in ['On rank 2 complex reflection groups', \emph{Comm. Algebra} \textbf{36}(6)
(2008), 2092-2132]. Virtual complex reflection groups generalise both complex
reflection groups of rank two and toric reflection groups, a family of groups
defined and studied by Gobet in ['Toric reflection groups', \emph{Journal of
the Australian Mathematical Society} \textbf{116} (2) (2024), 171-199]. We give
uniform presentations by generators and relations of virtual complex reflection
groups of rank two, which coincide with the presentations given by Brou\'e,
Malle and Rouquier in ['Complex reflection groups, braid groups, Hecke
algebras', \emph{J. reine angew. Math.} \textbf{500} (1998), 127-190.] when the
groups are finite. In particular, these presentations provide uniform
presentations for complex reflection groups of rank two where the generators
are reflections (however the proof uses the classification of irreducible
complex reflection groups). Moreover, we show that the center of virtual
complex reflection groups is cyclic, generalising what happens for irreducible
complex reflection groups of rank two and toric reflection groups. Finally, we
classify virtual complex reflection groups up to reflection isomorphisms.
