  This paper generalizes the method of deducing Dirac's equation to
constructing a family of equations that represent the $N$-th root of the basic
Energy-Momentum relation for a free particle \cite{Dattoli1, Dattoli2,
Dattoli3, Dattoli4}. Then these equations are recast in a form that allows one
to interpret them as the fundamental dynamical equations for particles with
fractional spin, which we study in detail for the case of $N=4$ and $N=3$. We
explicitly prove that the equation is invariant to rotations and boosts and
indeed represents spin-$\frac{3}{8}$ and spin-$\frac{1}{8}$ particles for $N=4$
and spin-$\frac{1}{6}$ and $0$ for $N=3$.
