Take the parametric solution for Pythagorean equation found by the Greeks, a=k*2pq, b=k*(p^2 - q^2), c= k*(p^2 + q^2) and substitute into the second equation.

p^2 - 2pq +3q^2 - 99/k = 0 

Solving this for p requires discriminant = 396/k - 8q^2 which will be a square.

So k=1,3,9,11,33,99 and checking each in turn gives these solutions:

   a       b       c

 27     120   123

 55     132   143

 72     135   153

 84     135   159

 99     132   165