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We calculate the Green's functions for the particle-vortex system, for twoanyons on a plane with and without a harmonic regulator and in a uniformmagnetic field. These Green's functions which describe scattering or boundstates (depending on the specific potential in each case) are obtained exactlyusing an algebraic method related to the SO(2,1) Lie group. From these Green'sfunctions we obtain the corresponding wave functions and for the bound stateswe also find the energy spectra.Comment: 21 Latex pages. Typos corrected. Results unchanged. Version to appear  in JM

Scattering and bound state Green’s functions on a plane via so(2,1) Lie algebra