  In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.
