The density property for Gizatullin surfaces of type $[[0,0,-r_2,-r_3]]$

Gizatullin surfaces 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 ${\mathbb{C}}_{x,y,u,v}^4$ 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.