For what the values of integer numbers k, m, n, a, b, c, d so the equation k/(x^2 + a x  + b) +  m/(x^2 + a x  + c) + n/(x^2 + a x  + d) = 0 have four integer solutions?

I find by my hand some equations have four integer solutions.

How can I tell Maple to do this? For what the values of integer numbers k, m, n, a, b, c, d so the equation
k/(x^2 + a x  + b) +  m/(x^2 + a x  + c) + n/(x^2 + a x  + d) = 0 have four integer solutions?