thus the system is uniform in $0 \le x \le L$. However, an important difference between them is that the boundary conditions for $\beta=0$ are Neumann, i.e., $\varphi'(0)=\varphi'(L)=0$, in contrast to the Dirichret ones for $\beta=1$. Thus, the eigenfunctions for $\beta=0$ and $1$ are described by cosine and sine functions, respectively. In terms of SUSY, the lowest eigenstate of $H_0$ is a singlet with zero energy, because it is annihilated by both $A_0$ and $A^\dagger_0$. Any other eigenstate of $H_0$ has the corresponding eigenstate of $H_1$, which give rise to a doublet of eigenstates with the same eigenvalue. Thus, we can express the plane-wave solutions for a ring of length $L$ by superposing the eigenfunctions of $H_0$ and $H_1$. We recall here that we have imposed periodic (antiperiodic) boundary conditions on $\varphi (x)$ depending on the total number of fermions $N$. Therefore, in Table I we should take the eigenfunctions in the row labeled by even (odd) $q$ when $N$ is even (odd). We also note that the boundary conditions for the eigenfunctions are indicated by P (periodic) and AP (antiperiodic) in Table I. In contrast to the cases of $\beta=0$ and $1$, the eigenfunctions (\[gegenbauer\]) with $\beta \ge 2$ 