| We consider a smooth affine variety V defined by an irreducible polynomial f(x, y, z) = x2 + y2 - xyz - z over C . We will describe all of its (rational) integral points via two automorphisms of V: one that is applied to the integral points of a twisted cubic curve lying on the surface with positive z-coordinates, and the other applied to one integral point on the surface with negative z-coordinate. Then we define the 'special' automorphism group of V to be the set of all automorphisms of V such that they can be lifted to automorphisms of A3 . Then we will show that the special automorphism group of V is a Coxeter group of rank 3. Further we will generalize the surface and compute its special automorphism group. |
