This paper provides a complete analytical solution to the Brownian motion of a self-propelled particle that moves in two spatial dimensions in the overdamped limit by satisfying kinematic constraints. The analytical expression of any-order moment of the probability distribution is obtained by a direct integration of the Langevin equation. Each moment is expressed as a multiple integral of the active motion performed by the particle. This allows obtaining any order moment for any active motion, i.e., when the particle is propelled by arbitrary time-dependent force and torque. For the special case when the ratio between the force and the torque is constant, the multiple integrals can be easily analytically solved and expressed as the real or the imaginary part of suitable analytic functions. As an application of the derived analytical results, the paper investigates the diusivity of the considered Brownian motion for constant and for arbitrary time-dependent force and torque.