Finding non-trivial integer solutions to x^3+y^3+z^3=1

So our instructor has given us a bonus question which is to find as many integer solutions to, $\displaystyle x^3+y^3+z^3=1$, such that $\displaystyle |x|,|y|,|z|>1$, as we can. He suggests that we use a computer algebra system to calculate the answers for us.

As I literally have zero experience with computational software I'm looking for advice on software, what I should try using, relevant guides, really anything anyone with even a slim iota of knowledge could offer as advice would be appreciated.

As background/research that I've already done;

I do have access to several computers.

I use linux.

I have not asked any of my programmer friends to "just do this for me".

The documentation for the two programs I've looked at, Sage and Maxima, makes it look relatively easy to solve certain problems, but don't appear to give info relevant to solving this problem.

But anyways since I know nothing really anything is helpful.

Thanks!

Re: Finding non-trivial integer solutions to x^3+y^3+z^3=1

Fermat Cubic.

We talked about solutions of the form $\displaystyle (x,y,z)=(1,n,-n)$ in the class and he didn't sound interested at all in solutions like that.

Fermat cubic - Wikipedia, the free encyclopedia

Chapter 4. n=x^3+y^3+z^3

It looks like I can use the parameterisation of the Fermat Cubic to generate integer answers?

Re: Finding non-trivial integer solutions to x^3+y^3+z^3=1

I'm thinking I can just arbitrarily select at least one of $(x,y,z)$ s.t. $x^3+y^3=1-z^3$. Say $z$ is fixed for some negative number then the RHS is a positive sum of two cubes. And it looks like there are more then a few lists of integer sequences, on oeis.org, that I can dig up numbers. I would have to compare them by hand though.