Let M be a 3-connected matroid, and let N be a 3-connected minor of M.
A pair {x1​,x2​}⊆E(M) is N-detachable if one of the matroids
M/x1​/x2​ or M\x1​\x2​ is both 3-connected and has an
N-minor. This is the second in a series of three papers where we describe the
structures that arise when it is not possible to find an N-detachable pair in
M. In the first paper in the series, we showed that if M has no
N-detachable pairs, then either M has one of three particular 3-separators
that can appear in a matroid with no N-detachable pairs, or there is a
3-separating set X with certain strong structural properties. In this paper,
we analyse matroids with such a structured set X, and prove that they have
either an N-detachable pair, or one of five particular 3-separators that can
appear in a matroid with no N-detachable pairs.Comment: 47 pages, 5 figure