Solutions to the diophantine equation $x^3+y^3+z^3+w^3=1$

Theorem.-The equation $x^3+y^3+z^3+w^3=n$ has an infinity of integer solutions if there exists one solution $(a,b,c,d)$ such that $$-(a+b)(c+d)\gt 0$$ is not a perfect square, and $a\ne b$, or $c\ne d$.

(See Diophantine equations, L. J. Mordell. Academic Press, p. 58).

Since $\mathbb F_7^3=\{0,\pm1\}$ necessarily one of the unknowns is a multiple of $7$ because if not we would have $$(\pm1)+(\pm1)+(\pm1)+(\pm1)\equiv 1\pmod7$$ which is impossible.

Trying to find a particular solution I find out $(x,y,z,w)=(14,\space 30,-23,-26)$ which satisfies the conditions of the above theorem. Thus, there are infinitely many solutions which are related, according to the mentioned book, to a Pell's equation (which as it is well known has an infinity of integer solutions).

In general it is clear that for such an equations there are infinitely many solutions. For example $$(n,-n,1,0),\; n\in \mathbb{Z}$$ and permutations. Another, less obvious, possibility is to consider the infinite family of solutions $$(9n^4,3n-9n^4,1-9n^3,0)\; n\in \mathbb{Z}$$ It could be interesting to know if all the solutions are contained in a finite number of curves or surfaces. For a similar problem: $$x^3+y^3+z^3=1$$ there is a nice discussion in the paper of D.H. Lehmer "On the diophantine equation $x^3+y^3+z^3=1$". I'm searching for an analogous paper in which is considered the case of four variables.