and $\phi$ tend asymptotically to finite values for $t\to \infty$. At infinity, the trajectory of the test particle is determined solely by the constant acceleration field and will thus be a parabola. The plane in which such asymptotic parabola lies is perpendicular to the $xy$ plane and its orientation is determined by the value to which the azimuthal angle $\phi$ tends asymptotically (which can be determined exactly by calculating the value of $\phi$ at the end of one period in fictitious time). This result could prove to be particularly useful in the design of powered planetary kicks (or flybys), a technique vastly used in modern interplanetary trajectory design [@danby_fundamentals_1988]. Planetary kicks are traditionally designed assuming an unperturbed hyperbolic motion around a certain planet. The outgoing conditions are then simply determined by the analytical expression governing Keplerian motion (i.e., a rotation of the hyperbolic access velocity). A different type of powered flyby can be considered, in which the spacecraft thrusts continuously in a fixed inertial direction. In such a case, and ignoring the fuel mass loss, the spacecraft conditions at infinity (i.e., when leaving the planet’s sphere of influence) can be determined exactly by a fully-analytical solution such as the one 