A new kind of conservation principle saying "Everything which is preserved by a dynamical system is also preserved by its associated torus actions" was proposed by Nguyen Tien Zung. 

In this talk, we will first review this conservation principle and its beautiful idea of the proof. Then we will see how the principle is confirmed in presence of singularities and apply it to normalize simultaneously a singular dynamical system together with its invariant singular geometric structures such as singular volume forms, singular  folded symplectic or log-symplectic forms, certain singular contact structures.

The talk is based on joint works with Tudor Ratiu and Nguyen Tien Zung.