[Math] How many cubes are the sum of three positive cubes

Question

Are there infinitely many integer positive cubes $x^3 = a^3 + b^3 + c^3$ that are equal to the sum of three integer positive cubes? If not, how many of them are there?

Answer 1

There are bivariate coprime polynomial parametrizations: https://sites.google.com/site/tpiezas/010

$$(a^4-2ab^3)^3 + (a^3 b+b^4)^3 + (2a^3 b-b^4)^3 = (a^4+a b^3)^3$$

Added If you drop the positivity constraint then there is another identity for $x=v^4$.

$$v^{12}=(v^4)^3=(9u^4)^3+(3uv^3-9u^4)^3+(v(v^3-9u^3))^3$$

I believe, but can't find it at the moment, that for all positive $x$ exist integers $a,b,c$ such that $x^3=a^3+b^3+c^3$.