Circular bidiagonal pairs

A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry is on the diagonal, or the subdiagonal, or in the top-right corner; (ii) each subdiagonal entry is nonzero, and the entry in the top-right corner is nonzero. Let $\mathbb F$ denote a field, and let $V$ denote a nonzero finite-dimensional vector space over $\mathbb F$. We consider an ordered pair of $\mathbb F$-linear maps $A: V \to V$ and $A^*: V \to V$ that satisfy the following two conditions: (i) there exists a basis for $V$ with respect to which the matrix representing $A$ is circular bidiagonal and the matrix representing $A^*$ is diagonal; (ii) there exists a basis for $V$ with respect to which the matrix representing $A^*$ is circular bidiagonal and the matrix representing $A$ is diagonal. We call such a pair a circular bidiagonal pair on $V$. We classify the circular bidiagonal pairs up to affine equivalence. There are two infinite families of solutions, which we describe in detail.