Space of initial values of a map with a quartic invariant

We compactify and regularize the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system turns out to have certain unusual properties, including a sequence of points of indeterminacy in $\mathbb P^1\cross \mathbb P^1$. These indeterminacy points are shown to lie on a singular fibre of the mapping to a corresponding QRT system and provide the existence of a one-parameter family of special solutions.