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In this article, we prove the existence of common fixed points for a pair ofmaps on a $q$-spherically complete $T_0$-ultra-quasi-metric space. The presentarticle is a generalization, in the assymmetric setting of the paper of Rao etal.\cite{Rao}. The key point in the proof is the use of Zorn's lemma. Weconstruct an appropriate chain and show that it has a maximal element, fromwhich we extract the fixed point we are looking for. The choice of the sets,here open balls, is characteristic of this type of problems and the contractioncondition are essential, specially when we are to establish the uniqueness ofthe fixed point

Common Coupled Fixed Point Theorems for Maps in a $T_0$-ultra-quasi-metric Space