# How to find the integer numbers a, b, c, m, n satifying the following conditions and remove the empty sets in this results?

How can I find the integer numbers a, b, c, m,n so that the funtion f[x]=(a x^2 + b x + c)/(m x + n) with the equation D[f[x], x]] == 0 have two integer solutions x1, x2 and the values f[x1] and f[x2] are also integer numbers?

I tried

Clear["Global*"]
f[x_] = (a x^2 + b x + c)/(m x + n);
x1 = (-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
x2 = (-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
list = Table[
If[GCD[a, b, c] == 1 && GCD[m, n] == 1 &&
a c m^2 - a b m n + a^2 n^2 > 0 && a n - b m  != 0 &&
a (c m^2 - b m n + a n^2) > 0 && b^2 - 4 * a * c < 0 &&
IntegerQ[x1] && IntegerQ[x2] && b c m n x1 x2 f[x1]* f[x2] != 0 &&
IntegerQ[f[x1]] &&
IntegerQ[f[x2]], {(a x^2 + b x  + c)/(m x + n), {x1, f[x1]}, {x2,
f[x2]}}, Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]


How can I remove the empty sets in my results?

Try

f[x_]:=(a x^2+b x+c)/(d x^2+e x+m);
FindInstance[{D[f[x1],x1]==0,D[f[x2],x2]==0,
0<a<=1,0<b<=1,0<c<=1,0<d<=1,0<e<=1,0<m<=1,
x1!=x2,Element[x1|x2,Integers]},{a,b,c,d,e,m,x1,x2}]


which finds an instance which seems to satisfy your conditions.

If you change all those <= to < then it finds a different solution.

I tried

Clear["Global*"]
f[x_] = (a x^2 + b x + c)/(m x + n);
x1 = (-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
x2 = (-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
list = Table[
If[GCD[a, b, c] == 1 && GCD[m, n] == 1 &&
a c m^2 - a b m n + a^2 n^2 > 0 && a n - b m  != 0 &&
a (c m^2 - b m n + a n^2) > 0 && b^2 - 4 * a * c < 0 &&
IntegerQ[x1] && IntegerQ[x2] && b c m n x1 x2 f[x1]* f[x2] != 0 &&
IntegerQ[f[x1]] &&
IntegerQ[f[x2]], {(a x^2 + b x  + c)/(m x + n), {x1, f[x1]}, {x2,
f[x2]}}, Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1,
10}, {n, 1, 10}];
Flatten[list, 4]


{{(7 + x + x^2)/(2 + x), {-5, -9}, {1, 3}}, {(10 + x + x^2)/( 3 + x), {-7, -13}, {1, 3}}, {(3 + 2 x + x^2)/( 1 + 2 x), {-2, -1}, {1, 2}}, {(5 + 2 x + x^2)/( 1 + x), {-3, -4}, {1, 4}}, {(7 + 2 x + x^2)/( 1 + 2 x), {-3, -2}, {2, 3}}, {(7 + 2 x + x^2)/( 3 + 2 x), {-4, -3}, {1, 2}}, {(10 + 2 x + x^2)/( 1 + x), {-4, -6}, {2, 6}}, {(3 + 3 x + x^2)/( 2 + x), {-3, -3}, {-1, 1}}, {(5 + 3 x + x^2)/( 4 + x), {-7, -11}, {-1, 1}}, {(6 + 3 x + x^2)/( 1 + x), {-3, -3}, {1, 5}}, {(6 + 3 x + x^2)/( 5 + x), {-9, -15}, {-1, 1}}, {(7 + 3 x + x^2)/( 6 + x), {-11, -19}, {-1, 1}}, {(8 + 3 x + x^2)/( 7 + x), {-13, -23}, {-1, 1}}, {(9 + 3 x + x^2)/( 8 + x), {-15, -27}, {-1, 1}}, {(10 + 3 x + x^2)/( 9 + x), {-17, -31}, {-1, 1}}, {(5 + 4 x + x^2)/( 2 + x), {-3, -2}, {-1, 2}}, {(6 + 4 x + x^2)/( 5 + 2 x), {-4, -2}, {-1, 1}}, {(7 + 4 x + x^2)/( 1 + x), {-3, -2}, {1, 6}}, {(7 + 4 x + x^2)/( 3 + x), {-5, -6}, {-1, 2}}, {(8 + 4 x + x^2)/( 1 + 2 x), {-3, -1}, {2, 4}}, {(8 + 4 x + x^2)/( 7 + 2 x), {-6, -4}, {-1, 1}}, {(10 + 4 x + x^2)/( 3 + 2 x), {-4, -2}, {1, 3}}, {(10 + 4 x + x^2)/( 9 + 2 x), {-8, -6}, {-1, 1}}, {(7 + 5 x + x^2)/( 2 + x), {-3, -1}, {-1, 3}}, {(7 + 5 x + x^2)/( 3 + x), {-4, -3}, {-2, 1}}, {(8 + 5 x + x^2)/( 1 + x), {-3, -1}, {1, 7}}, {(8 + 5 x + x^2)/( 4 + x), {-6, -7}, {-2, 1}}, {(10 + 5 x + x^2)/( 3 + x), {-5, -5}, {-1, 3}}, {(10 + 5 x + x^2)/( 6 + x), {-10, -15}, {-2, 1}}, {(10 + 6 x + x^2)/( 3 + x), {-4, -2}, {-2, 2}}, {(7 + x + 2 x^2)/( 1 + x), {-3, -11}, {1, 5}}, {(5 + 2 x + 2 x^2)/( 1 + 2 x), {-2, -3}, {1, 3}}, {(9 + 3 x + 2 x^2)/( 1 + x), {-3, -9}, {1, 7}}, {(4 + 5 x + 2 x^2)/( 2 + x), {-3, -7}, {-1, 1}}, {(5 + 5 x + 2 x^2)/( 3 + x), {-5, -15}, {-1, 1}}, {(6 + 5 x + 2 x^2)/( 4 + x), {-7, -23}, {-1, 1}}, {(7 + 5 x + 2 x^2)/( 5 + x), {-9, -31}, {-1, 1}}, {(8 + 5 x + 2 x^2)/( 6 + x), {-11, -39}, {-1, 1}}, {(9 + 5 x + 2 x^2)/( 7 + x), {-13, -47}, {-1, 1}}, {(10 + 5 x + 2 x^2)/( 8 + x), {-15, -55}, {-1, 1}}, {(5 + 6 x + 2 x^2)/( 3 + 2 x), {-2, -1}, {-1, 1}}, {(7 + 6 x + 2 x^2)/( 1 + 2 x), {-2, -1}, {1, 5}}, {(7 + 6 x + 2 x^2)/( 5 + 2 x), {-4, -5}, {-1, 1}}, {(9 + 6 x + 2 x^2)/( 7 + 2 x), {-6, -9}, {-1, 1}}, {(8 + 7 x + 2 x^2)/( 2 + x), {-3, -5}, {-1, 3}}, {(10 + x + 3 x^2)/( 1 + x), {-3, -17}, {1, 7}}, {(7 + 2 x + 3 x^2)/( 1 + 2 x), {-2, -5}, {1, 4}}, {(8 + 4 x + 3 x^2)/( 1 + 2 x), {-2, -4}, {1, 5}}, {(5 + 7 x + 3 x^2)/( 2 + x), {-3, -11}, {-1, 1}}, {(6 + 7 x + 3 x^2)/( 3 + x), {-5, -23}, {-1, 1}}, {(7 + 7 x + 3 x^2)/( 4 + x), {-7, -35}, {-1, 1}}, {(8 + 7 x + 3 x^2)/( 5 + x), {-9, -47}, {-1, 1}}, {(9 + 7 x + 3 x^2)/( 6 + x), {-11, -59}, {-1, 1}}, {(10 + 7 x + 3 x^2)/( 7 + x), {-13, -71}, {-1, 1}}, {(6 + 8 x + 3 x^2)/( 3 + 2 x), {-2, -2}, {-1, 1}}, {(7 + 8 x + 3 x^2)/( 2 + x), {-3, -10}, {-1, 2}}, {(8 + 8 x + 3 x^2)/( 5 + 2 x), {-4, -8}, {-1, 1}}, {(9 + 8 x + 3 x^2)/( 3 + x), {-5, -22}, {-1, 2}}, {(10 + 8 x + 3 x^2)/( 1 + 2 x), {-2, -2}, {1, 7}}, {(10 + 8 x + 3 x^2)/( 7 + 2 x), {-6, -14}, {-1, 1}}, {(9 + 10 x + 3 x^2)/( 3 + 2 x), {-2, -1}, {-1, 2}}, {(6 + 9 x + 4 x^2)/( 2 + x), {-3, -15}, {-1, 1}}, {(7 + 9 x + 4 x^2)/( 3 + x), {-5, -31}, {-1, 1}}, {(8 + 9 x + 4 x^2)/( 4 + x), {-7, -47}, {-1, 1}}, {(9 + 9 x + 4 x^2)/( 5 + x), {-9, -63}, {-1, 1}}, {(10 + 9 x + 4 x^2)/( 6 + x), {-11, -79}, {-1, 1}}, {(7 + 10 x + 4 x^2)/( 3 + 2 x), {-2, -3}, {-1, 1}}}