Suppose $a^2+b^2-c^2=0$ are formed by a (integral) Pythagorean triple. Then, there are $3\times3$ integer matrices to generate infinitely many more triples. [For example, take](https://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples#Pythagorean_triples_by_use_of_matrices_and_linear_transformations)
$$\begin{bmatrix}-1&2&2 \\ -2&1&2 \\ -2&2&3\end{bmatrix}\cdot
\begin{bmatrix}a\\ b\\ c\end{bmatrix}=\begin{bmatrix}u\\ v\\ w\end{bmatrix}.$$


One may wish to do the same with $a^3+b^3-c^3=0$, but Fermat's Last Theorem forbids it!


Alas! one settles for less $a^3+b^3-c^3=\pm1$. Here, we're in good company: $9^3+10^3-12^3=1$, coming from Ramanujan's taxicab number $1729=9^3+10^3=12^3+1^3$. There are plenty more.



> 
> **Question.** Does there exist a concrete $3\times3$ integer matrix $M=[m\_{ij}]$ such that whenever $a^3+b^3-c^3\in\{-1,1\}$ (integer tuple) then $u^3+v^3-w^3\in\{-1,1\}$ provided
>  $$\begin{bmatrix}m\_{11}&m\_{12}&m\_{13} \\ m\_{21}&m\_{22}&m\_{23} \\ m\_{31}&m\_{32}&m\_{33}\end{bmatrix}\cdot
> \begin{bmatrix}a\\ b\\ c\end{bmatrix}=\begin{bmatrix}u\\ v\\ w\end{bmatrix}.$$
> 
> 
> 



"Nearly" Fermat triples: case cubic
Any such matrix $M$ would give rise to an automorphism of the cubic surfaces
$$a^3 + b^3 = c^3 \pm d^3 \quad \subset \mathbb{P}^3.$$
These are both just different ways of writing the Fermat cubic surface
$$x\_0^3 + x\_1^3 + x\_2^3 + x\_3^3 = 0 \quad \subset \mathbb{P}^3.$$
It is well-known that the automorphism group of the Fermat cubic surface over $\mathbb{C}$ is generated by the "obvious automorphisms", namely by permuting coordinates and multiplying coordinates by a third root of unity (See Table 9.6 of "Dolgachev - Classical algebraic geometry" for this claim).


Hence, as you are interested with matrices with integer entries, we see that the only such $M$ are the 6 permutation matrices which permute $a,b,c$, with also possibly multiplying some variable by $-1$ to fix the signs.


For example
\begin{bmatrix}0&1&0 \\ 1&0&0 \\ 0&0&1\end{bmatrix}
is an example of such a matrix. The other matrices can be written down analogously.


