  We use a recently proposed scheme of matrix extension of dispersionless
integrable systems for the Abelian case, in which it leads to linear equations,
connected with the initial dispersionless system. In the examples considered,
these equations can be interpreted in terms of Abelian gauge fields on the
geometric background defined by the dispersionless system. They are also
connected with the linearisation of initial systems. We construct solutions to
these linear equations in terms of wave functions of the Lax pair for
dispersionless system, which is represented in terms of some vector fields.
