  We construct a universal local deformation (Kuranishi family) for pairs
consisting of a compact complex curve and a meromorphic 1-form.
  Each pair is assumed to be locally planar, a condition which in particular
forces the periods of the meromorphic differential to be preserved by local
deformations. The hyperelliptic case yields a universal local deformation for
the spectral data of integrable systems such as simply-periodic solutions of
the KdV equation or of the sinh-Gordon equation (cylinders of constant mean
curvature). This is the first of two papers in which we shall develop a
deformation theory of the spectral curve data of an integrable system.
