On the solution of the generalized Markov equation (Q1358496)

The paper announces some results about the solvability of the Diophantine equation \[ a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}+b_{1}x_{1}+ \cdots +b_{n}x_{n}+c=dx_{1} \cdots x_{n}, \] where \[ a_{1}, \cdots ,a_{n},d \in {\mathbb N},\quad b_{1}, \cdots ,b_{n},c \in {\mathbb N} \cup \{ 0 \}, \] \[ a_{1}\mid d,\cdots , a_{n}\mid d, a_{1}\mid b_{1}, \cdots ,a_{n}\mid b_{n},\quad n\geq 3. \] These results provide an algorithm for finding so-called fundamental solutions of the equation. From these solutions, all solutions of the equation may be obtained.