  Recently, V.Ginzburg introduced the notion of a principal nilpotent pair (=
pn-pair) in a semisimple Lie algebra {\frak g}. It is a double counterpart of
the notion of a regular nilpotent element. A pair (e_1,e_2) of commuting
nilpotent elements is called a pn-pair, if the dimension of their simultaneous
centralizer is equal to the rank of {\frak g} and some bi-homogeneity condition
is satisfied. Ginzburg proved that many familiar results of the `ordinary'
theory have analogues for pn-pairs. The aim of this article is to develop the
theory of nilpotent pairs a bit further and to present some applications of it
to dual pairs and sheets. It is shown that a large portion of Ginzburg's theory
can be extended to the pairs whose simultaneous centraliser is of dimension
rk{\frak g}+1. Such pairs are called almost pn-pairs. It is worth noting that
the very existence of almost pn-pairs is a purely "double" phenomenon, because
the dimension of "ordinary" orbits is always even. We prove that to any
principal or almost nilpotent pair one naturally associates a dual pair.
Moreover, this dual pair is reductive if and only if e_1 and e_2 can be
included in commuting sl_2-triples. We also study sheets containing members of
pn-pairs. Some cases are described, where these sheets are smooth and admit a
section.
