Recovering a local Lie group from structure constants

We construct a coordinate system of the 2nd kind corresponding to canonical coordinates of the 1st kind (in terminology of A. I. Maltsev), thereby obtaining a parametric solution of a Lie system of equations. We also give an integral representation of the group operations $f(x,y)$ of the local Lie group $G$ in canonical coordinates of the 1st kind. Our main tool is the modified formula of A. P. Yuzhakov for implicit mappings. The operation $f(x,y)$ is also represented as a power series, which is the reduced form of the Campbell–Hausdorff series.