Determine all positive integers $a$ for which the equation
$$\left ( 1 + \frac {1} {x} \right ) \cdot \left ( 1 + \frac {1} {x+1} \right ) \cdots \left ( 1 + \frac {1} {x+a} \right ) = a - x $$
has at least one integer solution $x$.

For each such integer $a$, determine the corresponding solutions.