Monogenic bialgebras over finite fields and rings of Witt vectors

We classify a certain collection of bialgebras that are generated by a single element over a finite field k, giving a parameterization using positive integers. We then put necessary and sufficient conditions on this pair so that the bialgebra lifts uniquely to the ring W(k) of Witt vectors with coefficients in k, and finally provide a formula for the number of such lifts.