Liouville Equation in 1/8 BPS Geometries

We investigate the 1/8 BPS geometries with SU(2) x U(1) x SO(4) x R symmetry in IIB supergravity which were classified by Gava et al, (hep-th/0611065). It is desirable to have a complete set of differential equations imposed on the controlling functions such that they are not only necessary but also sufficient to produce supergravity solutions with those symmetries. We work on this issue and find a new differential equation for the controlling functions. For a special case, we exhaust all the remaining constraints and show that they reduce to one Liouville equation. The solutions of this equation produce geometries which are locally equivalent to the near horizon geometries of intersecting D3-branes.