The author studies a class of Kähler metrics invariant under the group of holomorphic automorphisms of the Reinhardt domain \(\{(z,w):| z|^{2p}+| w|^ 2<1\}\), where \(z\in {\mathbb{C}}\), \(w\in {\mathbb{C}}^ n\), and \(p\neq 1\). Ordinary differential equations are derived that determine such metrics satisfying various conditions on the Ricci curvature, scalar curvature, or holomorphic sectional curvature.