This paper presents the HyDrone, a novel concept of hybrid vehicle for aerial and underwater applications. Here, we discuss its geometry and dynamical behavior, concerning their main advantages and constraints. Translation and rotation models in SE(3) are presented, with the respective control laws used to stabilize the system in the air and under the water. Similar to other air/water vehicles in the literature, our robot can also navigate in both environments without mechanical adaptation during the medium transitions. However, our vehicle provides better results when compared to other approaches in the state-of-the-art in terms of underwater navigation efficiency. Simulated results demonstrate the trade-offs associated with our new hybrid platform when compared to other existing hybrid vehicles.