  We consider the fermionic $R$-operator based on Bazhanov-Stroganov's
three-parameter elliptic parametrization of the free fermion model, and find
the most general solution of the related tetrahedral Zamolodchikov algebra in
the trigonometric limit for an arbitrary set of parameters. This allows to
construct an extension of the $R$-operator and the corresponding Yang-Baxter
equation, which are of the difference type in one of the spectral parameters.
