Is there software that helps list small solutions of the Diophantine equation $$ x_0^2=1+x_1^2+x_2^2+\cdots+ x_n^2 $$ where "small" is negotiable, but e.g. we could fix $x_0$ and and ask for the list of all solutions $x_1, \dots, x_n\in\mathbb Z$?

Have you had a look on this tutorial: http://reference.wolfram.com/mathematica/tutorial/DiophantineReduce.html ? To take different values n < n_max into account, a simple loop could work. 


This function in Mathematica find them all:
Invoke with:


