Found 837 solutions, though I might have missed a few small ones. The last was 2781957377^3-2781957376^3-2852818^3=825

This generates an infinite series of solutions that add to 2. See the special cases page. I need to filter these out to see the more interesting results more clearly. The same effect can be seen on all even contours, the nth contour generates an infinite sequence of solutions for n^3/4 (2→2, 4→16, 6→54, 8→128, 10→250, 12→432 and 14→686). This sequence continues forever, but the script only finds solutions up to 1000.

I ran this up to over 20 trillion, these are the only solutions where X is over a million that I found:

1101509^3-1101504^3-26304^3=701
3757419^3-3757414^3-59606^3=99
8756836^3-8756831^3-104776^3=289
4440347405^3+6662569^3-4440347410^3=134
76275664911^3+44356209^3-76275664916^3=64

Tested up to 1.17 trillion, solutions over 1 million (notice that 771 appears twice):

3077972^3-3077965^3-58378^3=771
14454639^3-14454632^3-163711^3=720
36460580^3-36460573^3-303358^3=771
89534194^3+552161^3-89534201^3=64
129629906^3-129629899^3-706659^3=538

I didn't find many solutions, I ran up to X = 129 billion, but the last solution found was for 125 at X = 196,416,133. Interestingly there are a lot of cubes, generally it seams that cubes are easier to find than other values.

1	(943  +  643  −  1033), (7293  −  7203  −  2423), (7293  +  2443  −  7383)
8	(1413  +  833  −  1503), (28403  −  28313  −  6013)
27	(463  −  373  −  363), (16523  +  4203  −  16613), (36303  −  36213  −  7083)
64	(1013  +  673  −  1103), (14473  −  14383  −  3833)
125	(1323  −  1233  −  763), (1964161333  −  1964161243  −  10136923)
216	(2583  +  1233  −  2673), (143633  +  17733  −  143723)
512	(26493  +  5753  −  26583), (140243  +  17453  −  140333)
1000	(272813  +  27193  −  272903)