The analytical expression for the propagation of Lissajous singularities through an astigmatic lens is derived and used to study the dynamic evolution of Lissajous singularities after passing through the astigmatic lens and the dependence of Lissajous singularities on the control parameters where the effect of astigmatism of the lens is stressed. It is shown that motion, pair creation and annihilation, handedness reversal and change in the degree of polarization of Lissajous singularities may take place. In particular, a single Lissajous singularity may appear and vanish due to the presence of the astigmatic lens.