A Model for the HyperKahler Metrics Building Program

Its by now well known that the harmonic superspace provides astrong framework for constructing general hyperKahler metrics. Origi-nal results in this issue are given by the Taub-Nut and Egushi-Hansonmetrics exhibiting both U(1) Pauli-Gursey isomertie as shown by Gib-bon et all in [1]. Its also shown that there exist series of potentialdepending explicitly on the harmonics variables u on the sphere S2.Based on these knowledge and on previous contributions to the pro-gram of metrics building [2, 3], we contribute by presenting a specialhyperKahler potential H+4 leading to an -parametrized Liouville equa-tion on the harmonic superspace.