The density property for Gizatullin surfaces completed by four rational curves

Gizatullin surfaces completed by a zigzag of type [[0,0,-r_2,-r_3]] can be described by the equations yu=xP(x), xv=uQ(u) and yv=P(x)Q(u) in \mathbb{C}^4_{x,y,u,v} where P and Q are non-constant polynomials. We establish the algebraic density property for smooth Gizatullin surfaces of this type. Moreover we also prove the density property for smooth surfaces given by these equations when P and Q are holomorphic functions.