Symmetry, Singularities, and Integrability in Complex Dynamics II.R escaling and Time-Translation in Two-Dimensional Systems

The explicit integrability of second-order ordinary differential equations invariant under time-translation and rescaling is investigated.Quadratic systems generated from the linearisable version of this class of equations are analysed to determine the relationship between the Painlev´e and singularity properties of the different systems.The transformation contains a parameter and for critical values, intimately related to the possession of the Painlev´e property in the parent second-order equation, one finds a difference from the generic behaviour.This study is a prelude to a full discussion of the class of transformations which preserve the Painlev´e property in the construction of quadratic systems from scalar nth-order ordinary differential equations invariant under time invariant under time translation and rescaling.