  We introduce a large class of Sobolev bi-orthogonal polynomial sequences
arising from a $LU$-factorizable moment matrix and associated with a suitable
measure matrix that characterizes the Sobolev bilinear form. A theory of
deformations of Sobolev bilinear forms is also proposed. We consider both
polynomial deformations and a class of transformations related to the action of
linear operators on the entries of a given bilinear form. Transformation
formulae among new and old polynomial sequences are determined. Finally,
integrable hierarchies of evolution equations arising from the factorization of
a time deformation of the moment matrix are presented.
