x2+y2=10, y2=(10-x2). Perfect squares of integers ≤10 are 0, 1, 4, 9.
Subtract each from 10: 10, 9, 6, 1. Of these only 9 and 1 are perfect squares (9=32 and (-3)2; and 1=12 and (-1)2. So x=±3, ±1 making y=±1, ±3.
(x,y)=(3,1), (3,-1), (-3, 1), (-3,-1), (1,3), (1,-3), (-1,3), (-1,-3) are the only integer solutions.