Smooth Sp(2,R)-actions on the 4-sphere

We construct a one-to-one correspondence between the equivariant diffeomorphism classes of smooth Sp(2, R)-actions on the standard 4-sphere without fixed points and the equivalence classes of certain pairs of R-actions and maps defined on the circle subject to five conditions. Consequently, we show that there are infinitely many smooth Sp(2, R)-actions on the space without fixed points up to equivariant diffeomorphisms.