On cohomogeneity one Hermitian non-Kähler metrics

We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries principal orbits codimension one. In particular, we focus on special class these constructed following B\'erard-Bergery which includes, among others, line bundles $\mathbb C\mathbb P^{m-1}$, linear Hopf and Hirzebruch surfaces. characterize their invariant metrics, such as balanced, K\"ahler-like, pluriclosed, locally conformally K\"ahler, Vaisman, Gauduchon. Furthermore, construct new examples cohomogeneity one metrics solving second-Chern-Einstein equation constant Chern-scalar curvature equation.