Symmetry, singularities and integrability in complex dynamics II: Rescaling 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 therelationship 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 nonlinear odes invariant under time translation and rescaling.