A counterexample to R. Davidson's conjecture on line processes

Abstract Rollo Davidson conjectured in 1968 that every stationary second order line process in the plane which has a.s. no parallel lines is necessarily a Cox (or doubly stochastic Poisson) process. This conjecture is disproved here. An affirmative answer is further given to the question whether there exists a lattice type point process in the plane which is stationary under arbitrary area preserving affine transformations.