  We analyse the structure of solutions of the Ginsparg-Wilson relation for
lattice Dirac operator in topologically trivial gauge sector. We show that the
properties of such solutions relating to the perturbative stability of the pole
of the fermion propagator as well as to the structure of the Yukawa models
based on these solutions are solely determined by the non-local chirally
invariant part of these Dirac operators. Depending on the structure of this
part, the pole in the fermion propagator may or may not be stable under
radiative corrections. We illustrate this by explicit examples.
