For fixed $n \in \mathbb{N}$ consider integer solutions to $$x^3+y^3+z^3=n \qquad (1) $$

If $n$ is a cube or twice a cube, identities exist.

Elkies suggests no other polynomial identities are known.

For which $n$ (1) has infinitely many integer solutions?

Added

Is there $n$, not a cube or twice a cube, which allows infinitely many solutions?

For fixed $n \in \mathbb{N}$ consider integer solutions to $$x^3+y^3+z^3=n \qquad (1) $$

If $n$ is a cube or twice a cube, identities exist.

Elkies suggests no other polynomial identities are known.

For which $n$ (1) has infinitely many integer solutions?

Added

Is there $n$, not a cube or twice a cube, which allows infinitely many solutions?

Added 2019-09-23:

The number of solutions can be unbounded.

For integers $n_0,A,B$ set $z=Ax+By$ and consider $x^3+y^3+(Ax+By)^3=n_0$. This is elliptic curve and it may have infinitely many rational points coming from the group law. Take $k$ rational points $(X_i/Z_i,Y_i/Z_i)$. Set $Z=\rm{lcm}\{Z_i\}$.

Then $n_0 Z^3$ has the $k$ integer solutions $(Z X_i/Z_i,Z Y_i/Z_i)$.

For fixed $n \in \mathbb{N}$ consider integer solutions to $$x^3+y^3+z^3=n \qquad (1) $$

If $n$ is a cube or twice a cube, identities exist.

Elkies suggests no other polynomial identities are known.

For which $n$ (1) has infinitely many integer solutions?

For fixed $n \in \mathbb{N}$ consider integer solutions to $$x^3+y^3+z^3=n \qquad (1) $$

If $n$ is a cube or twice a cube, identities exist.

Elkies suggests no other polynomial identities are known.

For which $n$ (1) has infinitely many integer solutions?

Added

Is there $n$, not a cube or twice a cube, which allows infinitely many solutions?