restart;
Vol:=(u,v,w,U,V,W)->sqrt(-u^2*U^4 + ((-W^2 + u^2 + v^2)*V^2 + (u^2 + w^2)*W^2 - u^4 + (v^2 + w^2)*u^2 - v^2*w^2)*U^2 - v^2*V^4 + ((v^2 + w^2)*W^2 + (v^2 - w^2)*u^2 - v^4 + v^2*w^2)*V^2 - W^2*(W^2*w^2 + (v^2 - w^2)*u^2 - v^2*w^2 + w^4))/12:
m:=20; nr:=0: nrmax:=50;
for U to m do
for V from U to m do
for W from V to U+V-1 do
for u to m do
for v from abs(W-u)+1 to W+u-1 do
for w from max(abs(W-u),abs(U-v))+1 to min(W+u,U+v)-1 do
vv:=Vol(u,v,w,U,V,W);
if vv::posint then nr++; lprint(nr, [u,v,w],[U,V,W],vol=vv) fi;
if nr>=nrmax then break 6 fi
od:od:od:od:od:od:
# Your example appears at the position 441 (for m=20)

m := 20

nrmax := 50

1, [8, 8, 7], [2, 3, 4], vol = 6

2, [17, 16, 16], [2, 3, 4], vol = 15

3, [6, 7, 8], [2, 4, 4], vol = 6

4, [6, 8, 7], [2, 4, 4], vol = 6

5, [7, 5, 6], [2, 4, 4], vol = 6

6, [7, 6, 5], [2, 4, 4], vol = 6

7, [4, 8, 7], [2, 5, 6], vol = 6

8, [7, 4, 4], [2, 5, 6], vol = 6

9, [16, 14, 13], [2, 5, 6], vol = 12

10, [4, 6, 5], [2, 7, 8], vol = 6

11, [6, 4, 4], [2, 7, 8], vol = 6

12, [8, 4, 3], [2, 7, 8], vol = 6

13, [12, 14, 13], [2, 7, 8], vol = 24

14, [10, 11, 12], [2, 8, 8], vol = 21

15, [10, 12, 11], [2, 8, 8], vol = 21

16, [11, 9, 10], [2, 8, 8], vol = 21

17, [11, 10, 9], [2, 8, 8], vol = 21

18, [17, 11, 12], [2, 8, 8], vol = 21

19, [17, 12, 11], [2, 8, 8], vol = 21

20, [8, 12, 11], [2, 9, 10], vol = 21

21, [11, 8, 8], [2, 9, 10], vol = 21

22, [8, 10, 9], [2, 11, 12], vol = 21

23, [10, 8, 8], [2, 11, 12], vol = 21

24, [17, 8, 8], [2, 11, 12], vol = 21

25, [12, 8, 7], [2, 13, 14], vol = 24

26, [16, 6, 5], [2, 13, 14], vol = 12

27, [16, 22, 21], [2, 13, 14], vol = 60

28, [17, 3, 4], [2, 16, 16], vol = 15

29, [17, 4, 3], [2, 16, 16], vol = 15

30, [12, 19, 19], [2, 19, 19], vol = 72

31, [8, 24, 23], [2, 19, 20], vol = 42

32, [17, 16, 17], [3, 5, 7], vol = 30

33, [4, 7, 8], [3, 6, 6], vol = 9

34, [4, 8, 7], [3, 6, 6], vol = 9

35, [14, 16, 17], [3, 6, 6], vol = 36

36, [14, 17, 16], [3, 6, 6], vol = 36

37, [4, 6, 6], [3, 7, 8], vol = 9

38, [7, 11, 10], [3, 7, 8], vol = 24

39, [8, 4, 2], [3, 7, 8], vol = 6

40, [11, 7, 8], [3, 7, 8], vol = 24

41, [16, 16, 17], [3, 7, 8], vol = 48

42, [16, 8, 10], [3, 8, 10], vol = 21

43, [11, 15, 16], [3, 9, 9], vol = 42

44, [11, 16, 15], [3, 9, 9], vol = 42

45, [7, 8, 7], [3, 10, 11], vol = 24

46, [9, 15, 14], [3, 11, 12], vol = 48

47, [8, 14, 13], [3, 13, 14], vol = 48

48, [9, 12, 11], [3, 14, 15], vol = 48

49, [9, 17, 15], [3, 15, 16], vol = 42

50, [11, 9, 9], [3, 15, 16], vol = 42