  Some consequences which follow from the periodicity assumption for spectral
density of Wilson--Dirac operator are studied. Such an assumption allows to
obtain simple representations for quark propagator, which reveals an important
role of $m\leftrightarrow -m$ symmetry. It is argued that this symmetry is
restored when the mirror fermion mass $m_r$ tended to infinity. The constrains
on zero modes of Wilson--Dirac operator in a toy model approximation are also
discussed.
