Given the path of a point particle, one can relate its acceleration and, in general, its kinematics to the curvature scalars of its trajectory. Using this, a general ansatz is made for the Yang-Mills connection corresponding to a non-Abelian point source. The Yang-Mills field equations are then solved outside the position of the point source under physically reasonable constraints such as finite total energy flux and finite total color charge. The solutions contain the Trautman solution; moreover two of them are exact whereas one of them is found using a series expansion in 1/R, where R is the retarded distance. These solutions are new and, in their most general form, are not gauge equivalent to the original Trautman solution.