Is it known whether the “sum of 3 cubes” problem has any parametric solution for integers other than 1 or 2 or multiples?

Question

For example, it is known that for all t, $$ (9t^4)^3 + (3t - 9t^4)^3 + (1 - 9t^3)^3 = 1 $$ and $$ (1 + 6t^3)^3 + (1 - 6t^3)^3 + (-6t^2)^3 = 2 $$

Answer 1

Cubic surface is rational. This means that there are 2-parametric parametrisation.

See this https://people.math.harvard.edu/~elkies/4cubes.html