A new method for generating measure invariant algorithms is presented. This method is based on a reformulation of the equations of molecular dynamics. These new equations are non-Hamiltonian but have a normal form which guarantees that the invariant measure is the canonical one for the new variables. Furthermore, from this normal form, one can easily build algorithms to integrate these equations. Using a Trotter-type factorization of the classical Liouville propagator, we build (time) reversible measure invariant integrators as successive direct translations. We apply this method to propose new algorithms to generate the Nosé–Hoover chain dynamics and the isothermal-isobaric dynamics. We also give a measure invariant integrator for the generalized Gaussian moment thermostating dynamics recently introduced by Liu and Tuckerman. Finally, we present numerical results which show comparable performances with previously proposed algorithms.