Generalised Manin transformations and QRT maps

Manin transformations are maps of the plane that preserve a pencil cubic curves. They composition two involutions. Each involution is constructed in terms an point required to be one base points pencil. We generalise this construction explicit birational quadratic resp. certain quartic pencils, and show they measure-preserving hence integrable. In multiplicity 2. On other hand, for pencils can any distinct (except points). employ Pascal's theorem admit infinitely many symmetries. The full 18-parameter QRT map obtained as special instance case limit where go infinity. by each generalised Manin transformation brought form fractional affine transformation. also specify classes which root.