  We extend the study of the inertial effects on the dynamics of active agents
to the case where self-alignment is present. In contrast with the most common
models of active particles, we find that self-alignment, which couples the
rotational dynamics to the translational one, produces unexpected and
non-trivial dynamics, already at the deterministic level. Examining first the
motion of a free particle, we contrast the role of inertia depending on the
sign of the self-aligning torque. When positive, inertia does not alter the
steady-state linear motion of an a-chiral self-propelled particle. On the
contrary, for a negative self-aligning torque, inertia leads to the
destabilization of the linear motion into a spontaneously broken chiral
symmetry orbiting dynamics. Adding an active torque, or bias, to the angular
dynamics the bifurcation becomes imperfect in favor of the chiral orientation
selected by the bias. In the case of a positive self-alignment, the interplay
of the active torque and inertia leads to the emergence, out of a saddle-node
bifurcation, of truly new solutions, which coexist with the simply biased
linear motion. In the context of a free particle, the rotational inertia leaves
unchanged the families of steady-state solutions but can modify their stability
properties. The situation is radically different when considering the case of a
collision with a wall, where a very singular oscillating dynamics takes place
which can only be captured if both translational and rotational inertia are
present.
