We investigate solutions of type II supergravity which have the product R-1
,R-3 x M-6 structure with non-compact M6 factor and which preserve at least
four supersymmetries. In particular, we consider various conifolds and the
N = 1 supersymmetric 'NS5-brane wrapped on a 2-sphere' solution recently d
iscussed in hep-th/0008001. In all of these cases, we explicitly construct
the complex structures, and the Kahler and parallel (3, 0)-forms of the cor
responding M-6. In addition, we verify that the above solutions preserve, r
espectively, eight and four supersymmetries of the underlying type II theor
y. We also demonstrate that the ordinary and fractional D3-brane (5-brane w
rapped on a 2cycle) solutions on singular, resolved and deformed conifolds,
and the (S-dual of) NS5-brane wrapped on 2-sphere can be obtained as speci
al cases from a universal ansatz for the supergravity fields, i.e. from a s
ingle one-dimensional action governing their radial evolution. We show that
like the 3-branes on conifolds, the NS5-brane on a 2-sphere background can
be found as a solution of a first-order system following from a superpoten
tial.