Diophantine equation problem

Shame they are going to close this, I actually know something about this one. Your form is equivalent to $$ X^2 + Y^2 + Z^2 + Y Z + Z X + X Y $$ meaning there is an invertible integral change of variables. This is a regular form and integrally represents the same numbers as $$ U^2 + V^2 + 2 W^2 $$ which is to say all numbers not of shape
$$ 4^k ( 16 m + 14 ) .$$ As it is positive definite, rather than just semidefinite, there are simple bounds on possible values for the variables in your original
$$ x^2 + y^2 + z^2 - y z - z x leq M$$
which can be derived either by eigenvalues for the related symmetric Gram matrix,or, as I do, by Lagrange multipliers in maximizing $x^2$ or $y^2$ or $z^2$ subject to the constraint given. So all solutions to $$ x^2 + y^2 + z^2 - y z - z x = M$$ can be found fairly quickly even for large $M.$

Well, see my article with Kaplansky and Schiemann,

http://zakuski.math.utsa.edu/~kap/Forms/Kap_Jagy_Schiemann_1997.pdf