Discrete Painlevé I and singularity confinement in projective space

We write the discrete Painlevé I equation as a polynomial map in projective space and follow the development of the singularity and the fate of the other initial value. Most of the time the initial value is at the next to leading term and we can recover it (and confine the singularity) at the (3m+1)th iteration by imposing a condition on the de-autonomization. For m=1 one gets the usual d-PI.