Congruences of clone lattices, II

We continue the study of congruences of clone lattices L-A, where A is finite, started in an earlier paper by the author and A. P. Semigrodskikh. We prove that each clone that either contains all unary operations or consists of essentially unary operations forms a one-element class of any non-trivial congruence of L-A. As a consequence, we get that L-A has the greatest non-trivial congruence provided the lattice is not simple, that L-A is directly indecomposable, and that it has neither distributive nor dually distributive elements except for the trivial ones. For |A|>2, no example of a non-trivial congruence is known so far. We exhibit some reasons why such congruences are not easy to find.