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Mignotte and Pethö used the Siegel-Baker method to find all the integral solutions (x,y,z) of the system of Diophantine equations x2-6y2=-5 and x=2z2-1. In this paper, we extend this result and put forward a generalized method which can completely solve the family of systems of Diophantine equations x2-6y2=-5 and x=az2-b for each pair of integral parameters a,b. The proof utilizes algebraic number theory and p-adic analysis which successfully avoid discussing the class number and factoring the ideals.