The first Painlevé equation $ $ P_I:y”=6y^2-x $ $ has the symmetries $ $ x \mapsto \omega x \y \mapsto \omega^3 y$ $ for any fifth root of unity $ \omega$ . At the same time, the near-infinity asymptotics of $ P_I$ involve the five rays $ $ \Gamma_k= \left\{x:\arg x= \frac{2 \pi i k}{5} \right\}, \quad k=0,1,2,3,4.$ $

Looking at the second Painlevé equation $ $ P_{II}: y”=2y^3+xy+\alpha $ $ the scaling symmetries are $ $ x \mapsto \omega x \ y \mapsto \omega^2 x $ $ for any third root of unity $ \omega$ . However, the near-infinity asymptotics of $ P_{II}$ actually involve six rays $ $ \Gamma_k= \left\{x:\arg x= \frac{2 \pi i k}{6} \right\}, \quad k=0,1,2,3,4,5.$ $ My question is: is there a direct relationship between the discrete symmetries of the second Painlevé equation, and its near-infinity asymptotics, as in the case of the first Painlevé equation? If so, how come I only found three symmetries, which should correspond to the rays $ $ \left\{x: \arg x=\frac{2 \pi i k}{3} \right\}, \quad k=0,1,2 $ $

Thanks!