Once you find one solution in integers, any other solution happens by adding an integer multiple of the coefficient cross product, namely
$$ \langle -16, 28, 31 \rangle $$
You know the solution $ \langle 7,6,5 \rangle \; . $
Any other integer solution is
$$ \langle 7-16t, 6+28t, 5+31t \rangle \; . $$
If $t > 0$ we get $7 - 16 t < 0.$ If $t < 0$ then $6 + 28 t < 0.$ It follows that $t=0,$ the only solution in positive integers is the given one. There are infinitely many integer solutions, they lie on the line I described, but that line passes only briefly through the first (positive) octant.