Geometrical loci and CFTs via the Virasoro symmetry of the mKdV-SG hierarchy: an excursus

We will describe the appearance of specific algebraic KdV potentials as a consequence of a requirement on an integro-differential expression. This expression belongs to a class generated by means of Virasoro vector fields acting on the KdV field. The “almost” rational KdV fields are described in terms of a geometrical locus of complex points. A class of solutions of this locus has recently appeared as a description of any conformal Verma module without degeneration.