  The invariance of the Lagrangian under time translations and rotations in
Kepler's problem yields the conservation laws related to the energy and angular
momentum. Noether's theorem reveals that these same symmetries furnish
generalized forms of the first integrals in a special nonconservative case,
which approximates various physical models. The system is perturbed by a
biparametric acceleration with components along the tangential and normal
directions. A similarity transformation reduces the biparametric disturbance to
a simpler uniparametric forcing along the velocity vector. The solvability
conditions of this new problem are discussed, and closed-form solutions for the
integrable cases are provided. Thanks to the conservation of a generalized
energy, the orbits are classified as elliptic, parabolic, and hyperbolic.
Keplerian orbits appear naturally as particular solutions to the problem. After
characterizing the orbits independently, a unified form of the solution is
built based on the Weierstrass elliptic functions. The new trajectories involve
fundamental curves such as cardioids and logarithmic, sinusoidal, and Cotes'
spirals. These orbits can represent the motion of particles perturbed by solar
radiation pressure, of spacecraft with continuous thrust propulsion, and some
instances of Schwarzschild geodesics. Finally, the problem is connected with
other known integrable systems in celestial mechanics.
