Period function and normalizers of vector fields in Rn with n−1 first integrals

We extend the theory that relates period function, normalizers and first integrals, which has been widely developed in R2, to any dimension. In particular, we show how to find n−1 independent normalizers of vector fields in Rn with n−1 independent first integrals. A formula for the derivatives of the period function is obtained and a necessary and sufficient condition for isochronicity is also proved. These results generalize previous works of Freire, Gasull and Guillamon, Sabatini and Villarini.