Implicit Ordinary or Partial Differential Equations have been widely studied
in recent times, essentially from the existence of solutions point of view.
One of the main issues is to select a meaningful solution among the infinitely
many ones. The most celebrated principle is the viscosity method. This
selection principle is well adapted to convex Hamiltonians, but it is not always
applicable to the non-convex setting.
In this work we present an alternative selecting principle that singles out the most regular solutions (which do not always coincide with the viscosity ones). Our method is based on a general regularity theorem for Implicit ODEs. We also provide several examples.