Symmetry in the Painlev\'e systems and their extensions to four-dimensional systems

We give a new approach to the symmetries of the Painlev'e equations $P_V,P_{IV},P_{III}$ and $P_{II}$, respectively. Moreover, we make natural extensions to fourth-order analogues for each of the Painlev'e equations $P_V$ and $P_{III}$, respectively, which are natural in the sense that they preserve the symmetries.