Towards a classification of two-character rational conformal field theories

We provide a simple and complete construction of infinite families of consistent, modular-covariant pairs of characters satisfying the basic requirements to describe two-character RCFT. These correspond to solutions of generic second-order modular linear differential equations. To find these solutions, we first construct "quasi-characters" from the Kaneko-Zagier equation and subsequent works by Kaneko and collaborators, together with coset dual generalisations that we provide in this paper. We relate our construction to the Hecke images recently discussed by Harvey and Wu.