positive mass and the other two with negative mass. These two branches correspond to $+1$ and $-1$ eigenvalues of $\gamma_{5}$ in the taste space. The latter splits them into one with positive mass, two with zero mass and the other one with negative mass. We note that $M_{A}$ and $M_{H}$ are also derived from the flavored mass terms for naive fermions through spin-diagonalization as shown in [@CKM1]. Now we introduce a Wilson parameter $r=r\delta_{x,y}$ and shift mass as in Wilson fermions [@Hoel]. Then the Adams-type staggered-Wilson fermion action is given by $$\begin{aligned}
S_{\rm A}\,&=\,\sum_{xy}\bar{\chi}_{x}[\eta_{\mu}D_{\mu}
+r(1+M_A)+M]_{xy}\chi_{y}
\ ,
\label{AdamsS}
\\
D_{\mu}&={1\over{2}}(V_{\mu}-V^\dagger_{\mu}) 
\ .\end{aligned}$$ Here $M$ stands for the usual taste-singlet mass ($M=M\delta_{x,y}$). The Hoelbling-type fermion action is given by $$\begin{aligned}
S_{\rm H}\,=\,\sum_{xy}\bar{\chi}_{x}[\eta_{\mu}D_{\mu}
+r(2+M_H)+M]_{xy}\chi_{y}
\ .
\label{HoelS}\end{aligned}$$ It is obvious that these lattice fermions have possibility to be two- or one-flavor Wilson fermions. In lattice QCD simulation with these fermions, the mass parameter $M$ will be tuned to take a chiral limit as Wilson fermions. For some negative value of the mass parameter: $-1<M<0$ for Adams-type and $-2<M<0$ for Hoelbling-type, we obtain two-flavor and one-flavor overlap fermions respectively by using the overlap formula in principle.

Discrete Symmetries {#sec:Sym}
===================

In this section, 