  We show that the average number of integral points on elliptic curves,
counted modulo the natural involution on a punctured elliptic curve, is bounded
from above by $2.1 \times 10^8$. To prove it, we design a descent map, whose
prototype goes at least back to Mordell, which associates a pair of binary
forms to an integral point on an elliptic curve. Other ingredients of the proof
include the upper bounds for the number of solutions of a Thue equation by
Evertse and Akhtari-Okazaki, and the estimation of the number of binary quartic
forms by Bhargava-Shankar. Our method applies to $S$-integral points to some
extent, although our present knowledge is insufficient to deduce an upper bound
for the average number of them. We work out the numerical example with
$S=\{2\}$.
