Systems of first order ordinary differential equations allowing a given 3-dimensional Lie group as a subgroup of their symmetry group

We determine systems of the first order ordinary differential equations such that their group of symmetries contains a three-dimensional Lie subgroup G. We represent the basis vectors of the Lie algebra \(\mathfrak {g}\) of G by vector fields in the three-dimensional real space. Two cases are distinguished according to whether the infinitesimal generators of \(\mathfrak {g}\) do not contain any component or contain component with respect to the independent variable of the system.