Supergravity solutions without triholomorphicU(1)isometries

We investigate the construction of five-dimensional supergravity solutions that do not have any triholomorphic $U(1)$ 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-Kähler base spaces that can be put into a Gibbons-Hawking form and hyper-Kähler base spaces that cannot be put into a Gibbons-Hawking form. We find a new solution which is over triaxial Bianchi type IX Einstein–hyper-Kähler base space with no triholomorphic $U(1)$ symmetry. One special case of this solution corresponds to a five-dimensional solution based on Eguchi-Hanson type II geometry.