Tackling Markoff-Hurwitz Equations

We present algorithms for searching and generating solutions to the equation x
₁
²
+x
₂
²
+ ...+x
_n
²
= kx
₁
x
₂
...x
_n
. Solutions are reported for n = 2, 3,..., 9. Properties of solutions are discussed. We can prove that the solutions do not exist when n=4 and k=2 or 3, n=5 and k=2 or 3. Conjectures based on computational results are discussed.