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We present most general one-parametric solutions of the Yang-Baxter equations(YBE) for one spectral parameter dependent $R_{ij}(u)$-matrices of the six- andeight-vertex models, where the only constraint is the particle numberconservation by mod(2). A complete classification of the solutions isperformed. We have obtained also two spectral parameter dependent particularsolutions $R_{ij}(u,v)$ of YBE. The application of the non-homogeneoussolutions to construction of Zamolodchikov's tetrahedral algebra is discussed.

On the solutions of the Yang-Baxter equations with general inhomogeneous eight-vertex $R$-matrix: Relations with Zamolodchikov's tetrahedral algebra