  The classical Pell equation can be extended to the cubic case considering the
elements of norm one in $Z[\sqrt[3]{r}]$, which satisfy $x^3 + r y^3 + r^2 z^3
- 3 r x y z = 1$. The solution of the cubic Pell equation is harder than the
classical case, indeed a method for solving it as Diophantine equation is still
missing. In this paper, we study the cubic Pell equation over finite fields,
extending the results that hold for the classical one. In particular, we
provide a novel method for counting the number of solutions in all possible
cases depending on the value of r. Moreover, we are also able to provide a
method for generating all the solutions.
