On some linear equations associated with dispersionless integrable systems

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