Explicit Calabi–Yau metrics in D=6D=6 possessing an isometry group with orbits of codimension one

A method for constructing explicit Calabi–Yau metrics in six dimensions with an isometry group with orbits of codimension one is presented. The equations to solve are nonlinear, but become linear when certain geometrical objects defining the metric vary over a complex submanifold. It is shown that this method encode known examples such as the CY metrics of [G. Gibbons, H. Lu, C. Pope, K. Stelle, Nuclear Phys. B 623 (2002) 3] or the asymptotic form of the BKTY metrics of [S. Bando, R. Kobayashi, Math. Ann. 287 (1990) 175; Proc. 21st Int. Taniguchi Symp, Lecture Notes in Pure Math. 1339 (1988) 20] and [G. Gibbons, P. Rychenkova, J. Geom. Phys. 32 (2000) 311], but we construct new ones.