First integrals for the Kepler problem with linear drag

In this work we consider the Kepler problem with linear drag, and prove the existence of a continuous vector-valued first integral, obtained taking the limit as (trightarrow +infty ) of the Runge–Lenz vector. The norm of this first integral can be interpreted as an asymptotic eccentricity (e_{infty }) with (0le e_{infty } le 1). The orbits satisfying (e_{infty } <1) approach the singularity by an elliptic spiral and the corresponding solutions (x(t)=r(t)e^{itheta (t)}) have a norm r(t) that goes to zero like a negative exponential and an argument (theta (t)) that goes to infinity like a positive exponential. In particular, the difference between consecutive times of passage through the pericenter, say (T_{n+1} -T_n), goes to zero as (frac{1}{n}).