User blog comment:DrCocktor/Cubes in the form 1 + k + k^2/@comment-35470197-20190413133559

Although I do not know the answer, your question is equivalent to the equation "Is there any solution (a,b) of the equality a^3-3ab^2+b^3 = 1 other than (2,-1) and (-3,-2)?" because Z[(1+√-3)/2] is PID. I do not know whether this interpretation reduces the cost to find solutions or not.