  In a previous paper we used T-duality to construct a new type of 1/4-BPS
solution describing a pair of NS5-branes intersecting in 1+3 dimensions and
localised in all other directions except for a single transverse circle. This
led to an explicit solution to a sourced Monge--Ampere equation, of which there
are few known examples. In this paper we refine this formalism and apply it to
two important generalisations: the resolved and deformed conifolds. In doing so
we construct two new solutions describing, respectively, a pair of NS5-branes
separated in a transverse direction and a pair of NS5-branes with smooth
`diamond' profile. We show how the parameter of the resolved conifold (size of
the S^2) maps to a transverse separation of the NS5-branes, while the modulus
of the deformed conifold (size of the S^3) maps to the deformation parameter of
the diamond web.
