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 $(Z X_i/Z_i,Z Y_i/Z_i)$.

all$x, y, z$ are positive, though I think this may serve as a little bit of help: $$\begin{align} 1 &= (9t^3 + 1)^3 + (9t^4)^3 + (-9t^4 - 3t)^3 \\ 2 &= (6t^3 + 1)^3 + (-6t^3 - 1)^3 + (-6t^2)^3 \end{align}$$ or forbigsolutions for $n = 1$: $$1 = (1 - 9t^3 + 648t^6 + 3888t^9)^3 + (-135t^4 + 3888t^{10})^3 + (3t - 81t^4 - 1296t^7 - 3888t^{10})^3$$ $\endgroup$ – Mr Pie Oct 27 '17 at 5:38