Undefined Jacobi last multiplier? Complete symmetry group!

The Jacobi last multiplier has an intimate association with the symmetries of differential equations. We present four examples in which the inability to obtain a last multiplier using the method of Lie itself provides us with information.We show that the three representations of the complete symmetry group of the linear harmonic oscillator can be obtained by searching for Jacobi last multipliers: they correspond to the Lie point symmetries that have zero determinant, thus far regarded as a useless case. The point is emphasized using the examples of the Volterra–Verhulst–Pearl equation, the Kepler problem and a scalar equation of fourth order.