Fundamental domain for the Markoff-Hurwitz equation

For integers $a\mathrm{\ne }0$, $k$, and $n\ge 3$, we consider the Markoff-Hurwitz equation given by $x_1^1+\cdots +x_n^2-a{x}_1\cdots x_n=k$. By defining graphs associated with a height function and by using their properties, we find an exact fundamental domain for a symmetric group generated by involution maps sending $(x_1,\dots ,x_n)$ to $(x_1,\dots ,a{x}_1\cdots x_{i-1}{x}_{i+1}\cdots x_n-x_i,\dots ,x_n)$, permutations, and double sign changes on the set of integral solutions for the Markoff-Hurwitz equation.