I was looking for integer solutions to this equation: $$a^3+b^3+c^3-3abc=d^3$$ And found a parametric solution. Given u, v, w : \begin{cases} a=3\left(u^2v+v^2w+w^2u\right)\\ b=3\left(uv^2+vw^2+wu^2\right)\\ c=u^3+v^3+w^3+6uvw\\ d=u^3+v^3+w^3-3uvw \end{cases}
A Natural Extension of the Pythagorean Equation to Higher Dimensions http://www.math.grinnell.edu/~chamberl/papers/pythagorean.pdf
Above equation shown below:
$\begin{cases} a=3\left(u^2v+v^2w+w^2u\right)\\ b=3\left(uv^2+vw^2+wu^2\right)\\ c=u^3+v^3+w^3+6uvw\\ d=u^3+v^3+w^3-3uvw \end{cases}$
$(a,b,c,d)=(2,4,3,3)$
Above value's is not a solution for the equation (given in bracket).
Since the equation submitted by "OP" does not generate all possible
solutions the answer to his question is negative.