a simple Lie algebra of classical type and semisimple

. We complete the classification of conformal embeddings of a maximally reductive subalgebra k into a simple Lie algebra g at non-integrable non-critical levels k by dealing with the case when k has rank less than that of g . We describe some remarkable instances of decomposition of the vertex algebra V k ( g ) as a module for the vertex subalgebra generated by k . We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings A 1 × A 1 ֒ → C 3 at level k = − 1 / 2, and obtain explicit branching rules by applying certain q -series identity. In the analysis of conformal embedding A 1 × D 4 ֒ → C 8 at level k = − 1 / 2 we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.