Abstract
Let HCI be the Hua construction of the first type. We describe the Einstein-Kahler metric for HCI. We reduce the Monge-Ampe re equation for the metric to an ordinary differential equation in the auxiliary function\(X\left( {z,w,\zeta } \right)\) This differential equation can be solved to give an implicit function in\(X\left( {z,w,\zeta } \right)\). For some cases, we obtained the solution of the differential equation and the explicit forms of the complete Einstein-Kahler metrics on HCI which are the non-homogeneous domains.
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