  The ground state of the antiferromagnetic XY model with a kagome lattice is
characterized by a well developed accidental degeneracy. As a consequence the
phase transition in this system consists in unbinding of pairs of fractional
vortices. Addition of the next-to-nearest neighbors (NNN) interaction leads to
stabilization of the long-range order in chirality (staggered chirality). We
show that the phase transition, related with destruction of this long-range
order, can happen as a separate phase transition below the temperature of the
fractional vortex pairs unbinding only if the NNN coupling is extremely weak,
and find how the temperature of this transition depends on coupling constants.
We also demonstarte that the antiferromagnetic ordering of chiralities and,
accordingly, the presence of the second phase transition are induced by the
free energy of spin wave fluctuations even in absence of the NNN coupling.
