  The momentum space zero-range model is used to investigate universal
properties of three interacting particles confined to two dimensions. The
pertinent equations are first formulated for a system of two identical and one
distinct particle and the two different two-body subsystems are characterized
by two-body energies and masses. The three-body energy in units of one of the
two-body energies is a universal function of the other two-body energy and the
mass ratio. We derive convenient analytical formulae for calculations of the
three-body energy as function of these two independent parameters and exhibit
the results as universal curves. In particular, we show that the three-body
system can have any number of stable bound states. When the mass ratio of the
distinct to identical particles is greater than 0.22 we find that at most two
stable bound states exist, while for two heavy and one light mass an increasing
number of bound states is possible. The specific number of stable bound states
depends on the ratio of two-body bound state energies and on the mass ratio and
we map out an energy-mass phase-diagram of the number of stable bound states.
Realizable systems of both fermions and bosons are discussed in this framework.
