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

Universal Deformation of a Curve and a Differential