Complete systems of differential invariants of vector fields in a euclidean space

The system of generators of the differential field of all G-invariant differential rational functions of a vector field in the n-dimensional Euclidean space R^n is described for groups G = M(n) and G = SM(n), where M(n) is the group of all isometries of Rn and SM(n) is the group of all euclidean motions of R^n . Using these results, vector field analogues of the first part of the Bonnet theorem for groups Aff(n), M(n), SM(n) in R^n are obtained, where Aff(n) is the group of all affine transformations of R^n. These analogues are given in terms of the first fundamental form and Christoffel symbols of a vector field.