  This is the second in a series of papers on Poisson formalism for the cubic
nonlinear Schr\"{o}dinger equation with repulsive nonlinearity. In this paper
we consider periodic potentials. The inverse spectral problem for the periodic
auxiliary Dirac operator leads to a hyperelliptic Riemann surface $\G$. Using
the spectral problem we introduce on this Riemann surface a meromorphic
function $\P$. We call it the Weyl function, since it is closely related to the
classical Weyl function discussed in the first paper. We show that the pair
$(\G,\P)$ carries a natural Poisson structure. We call it the deformed
Atiyah--Hitchin bracket. The Poisson bracket on the phase space is the image of
the deformed Atiyah--Hitchin bracket under the inverse spectral transform.
