If you want solution for this precise equation, look for "thue
equation".
The thue equations are some of the few for which there exists
efficient methods.
for example in PARI/GP (from sage with gp_console())
sage: gp_console()
GP/PARI CALCULATOR Version 2.3.3 (released)
[snip]
PARI/GP is free software, covered by the GNU General Public License,
and
comes WITHOUT ANY WARRANTY WHATSOEVER.
Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.
parisize = 8000000, primelimit = 500000
? t = thueinit(x^3-3*x-1) \\ corresponds to the homogenized
x^3-3*x*y^2-y^3
? thue(t,1) \\ solve with rhs = 1
%2 = [[-3, 2], [-1, 1], [0, -1], [1, -3], [2, 1], [1, 0]]
PS: this might be wrapped in SAGE, I didn't checked.
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