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

Abstract 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 $$\mathfrak {g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> of G by vector fields in the three-dimensional real space. Two cases are distinguished according to whether the infinitesimal generators of $$\mathfrak {g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> do not contain any component or contain component with respect to the independent variable of the system.