Sharp bounds for the number of integral points on $y^2=x^3\pm 2tx^2+tpx$

In this paper, we consider a family of elliptic curves over the rational numbers which arise from a well-known problem of Zagier. The main result is to prove a sharp upper bound of one nontrivial integral point on any curve in this family. This result generalizes some results which appear in the literature, and simplifies the proof for those previously known cases.