Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces

We describe the set of all locally nilpotent derivations of the quotient ring K[X,Y,Z]/(f(X)Y−φ(X,Z))K[X,Y,Z]/(f(X)Y−φ(X,Z)) constructed from the defining equation f(X)Y=φ(X,Z)f(X)Y=φ(X,Z) of a generalized Danielewski surface   in K3K3 for a specific choice of polynomials f and φ  , with KK an algebraically closed field of characteristic zero. As a consequence of this description we calculate the ML  -invariant and the Derksen invariant of this ring. We also determine a set of generators for the group of KK-automorphisms of K[X,Y,Z]/(f(X)Y−φ(Z))K[X,Y,Z]/(f(X)Y−φ(Z)) also for a specific choice of polynomials f and φ.