Integral points on a family of elliptic curves

We consider the family of elliptic curves with equation [ $y^{2}+y=x^{3}-a^{2x}$ ] Using various Diophantine tools (reduction mod p, Néron-Tate heights, linear forms in elliptic logarithms) we will show many properties of the set of integral points and especially related the set to the rank of the elliptic curve.