An explicit example of an iteration group in the ring of formal power series

We give an example of some iteration group in a ring of formal power series over a field of characteristic 0. It allows us to obtain an explicit formula for some one-parameter group of (truncated) formal power series under an additional condition. Consequently, we are able to show some non-commutative groups of solutions of the third Aczél-Jabotinsky differential equation in the ring of truncated formal power series.