On the geometry of a 4-dimensional extension of a q-Painlev'e I equation with symmetry type A1 (1)

Abstract We present a geometric study of a four-dimensional integrable discrete dynamical system which extends the autonomous form of a q-Painlevé I equation with symmetry of type A₁^ (1). By resolution of singularities it is lifted to a pseudo-automorphism of a rational variety obtained from (P¹) ^ 4 by blowing up along 28 subvarieties and we use this to establish its integrability in terms of conserved quantities and degree growth. We embed this rational variety into a family which admits an action of the extended affine Weyl group W (A₁^ (1) ) W (A₁^ (1) ) by pseudo-isomorphisms. We use this to construct two 4-dimensional analogues of q-Painlevé equations, one of which is a deautonomisation of the original autonomous integrable map.