  We investigate the construction of five-dimensional supergravity solutions
that don't have any tri-holomorphic U(1) isometries. We construct a class of
solutions that in various limits of parameters reduces to many of previously
constructed five-dimensional supergravity solutions based on both hyper-Kahler
base spaces that can be put into a Gibbons-Hawking form and hyper-Kahler base
spaces that can't be put into a Gibbons-Hawking form. We find a new solution
which is over triaxial Bianchi type IX Einstein-hyperkahler base space with no
tri-holomorphic U(1) symmetry. One special case of this solution corresponds to
five-dimensional solution based on Eguchi-Hanson type II geometry.
