Geometric bounds on the linearization domain and analytic dependence on parameters for families of analytic vector fields in a neighborhood ofa singular point

We study families of holomorphic vector fields, holomorphically depending on parameters,in a neighborhood of an isolated singular point. When the singular point is in the Poincarédomain for every vector field of the family we prove, through a modification of classicalSternberg’s linearization argument, cf. Nelson (1969) [7] too, analytic dependence onparameters of the linearizing maps and geometric bounds on the linearization domain:each vector field of the family is linearizable inside the smallest Euclidean sphere which isnot transverse to the vector field, cf. Brushlinskaya (1971) [2], Ilyashenko and Yakovenko(2008) [5] for related results. We also prove, developing ideas in Martinet (1980) [6],a version of Brjuno’s Theorem in the case of linearization of families of vector fields neara singular point of Siegel type, and apply it to study some 1-parameter families of vectorfields in two dimensions.