Consider an instance of the shifted Lonely Runner Conjecture (shifted-LRC) where n runners (except the stationary runner 0) have integer speeds and start from real values in [0,1[ at time t=0. We show that one can derive an alternative vector of starting points that can be made to be arbitrarily close to the initial vector of starting points. The alternative starting point of each runner i is a rational in [0,1[ and is expressible as (qi / P) where P is a large prime and qi is an integer in [0, P-1]. We then introduce a new LRC variant called the equi-split-LRC. The LRC instance allows a minimal loneliness gap of f for runner 0 from the remaining n-1 runners, if and only if, the equi-split-LRC instance with the same vector of speeds and alternative vector of starting points simultaneously allows a minimal loneliness gap of f/P for the arc-center of each of P sectors into which the circle is divided from the remaining n-1 runners. Here, f is a desired fraction in ]0,1[. This finding is important in the light of recent counter-examples to the shifted-LRC.