Complete solutions of the simultaneous Pell’s equations (a²+4)x²－y²=4 and x²－bz²=1

That the necessary and sufficient conditions that the simultaneous Pell equations (a²+4)x²－y²=4 and x²－bz²=1 have positive integer solutions are obtained by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lehmer sequence and the associated Lehmer sequence. Moreover, that these simultaneous Pell equations have at most one solution in positive integers are proved. When a solution exists, the only solution of the system is given by the cube or 5^th power of the minimal positive integer solution of the first equation of the title equations except for the three special solutions.