Cubic surfaces in affine three space tend to have few integral points . Certain cubics such as x^3+y^3+z^3=m ,may have many such points but very little is known about them .We discuss these questions for Markoff type surfaces : x^2+y^2+z^2-x.y.z= m , for which a (nonlinear) descent is a starting point to investigate the Hasse Principle and strong approximation ,and also "class numbers" and their averages for the corresponding group of polynomial morphisms of affine three space.

Joint work with Bourgain / Gamburd and Ghosh.