An algebraic structure of groups of solutions of the third Aczel-Jabotinsky formal differential equation

We study the algebraic structure of the groups of solutions of the third Acz{é}l-Jabotinsky differential equation $(H\circ \mathrm{\Phi })(x)=\frac{d\mathrm{\Phi }}{dx}\cdot H(x)$ in the rings of formal power series and truncated formal power series $k[[x]{]}_s$, where $s$ is either a positive integer or $s=\mathrm{\infty }$ and $H\in k[[x]{]}_s$ is given. We give also the detailed description of all solutions of this equation.