  We study deterministic dynamics of overactive Brownian particles in 2D and 3D
potentials. This dynamics is Hamiltonian. Integrals of motion for continuous
rotational symmetries are reported. The cases of 2D, axisymmetric and
non-axisymmetric 3D potentials are characterized and compared to each other.
The strong impact of the rotational symmetry integrals of motion on the chaotic
and quasiperiodic orbits is revealed. The scattering cross section is reported
for spherically symmetric Gaussian-shape potential wells and obstacles as a
function of self-propulsion speed.
