Can you describe, in parametric form or in any other explicit way, all rational solutions to any of the following equations: $$ y^2 + z^2 = x^3+1, $$ $$ y^2 + z^2 = x^3-1, $$ $$ y^2+x^2y+z^2+1=0. $$ Please consider each equation separately, this is not a system.

For the first two equations, I earlier asked the same question for integer solutions, see How to describe all integer solutions to $x^2+y^2=z^3+1$? , and no way more explicit than try all $x$ and solve the resulting equation in $(y,z)$ has been suggested. However, sometimes rational solutions are easier to parametrize than integer ones.

The last equation has no integer solutions, but has rational ones.

The motivation for this question is that there three equations are the smallest ones (in a sense defined here: What is the smallest unsolved Diophantine equation?) for which the problem of describing all rational solutions is non-trivial.

Of course, the problem is equivalent to describing all coprime integers (or all integers, or all rationals) solutions to the homogeneous equations $$ y^2t + z^2t = x^3+t^3, $$ $$ y^2t + z^2t = x^3-t^3, $$ $$ y^2t+x^2y+z^2t+t^3=0, $$ respectively.