  We establish a one-to-one correspondence between conjugacy classes of any
Hecke group and irreducible systems of poles of rational period functions for
automorphic integrals on the same group. We use this correspondence to
construct irreducible systems of poles and to count poles. We characterize
Hecke-conjugation and Hecke-symmetry for poles of rational period functions in
terms of the transpose of matrices in conjugacy classes. We construct new
rational period functions and families of rational period functions.
