  This is the first in a series of papers on Poisson formalism for the cubic
nonlinear Schr\"{o}dinger equation with repulsive nonlinearity and its relation
to complex geometry. In this paper we study general continuous potentials. We
demonstrate that the Weyl functions of the corresponding auxiliary Dirac
spectral problem carry a natural Poisson structure. We call it the
Atiyah--Hitchin Poisson bracket. We show that the Poisson bracket on the phase
space is the image of the Atiyah--Hitchin bracket on Weyl functions under the
inverse spectral transform.
