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We give a parametrization of zero-dimensional ideals in the power series ring $k[[x,y]]$ with a given leading term ideal with respect to local lex ordering in terms of certain canonical Hilbert-Burch matrices. This is an extension to the local setting of the parametrizations of Gröbner cells obtained in the polynomial ring $k[x,y]$ by Conca and Valla for lex ordering and Constantinescu for deglex.