This paper presents a new form of the Yang–Baxter equation (YBE) for R-matrices that depend on three spectral parameters. An R-matrix is the local operator that builds transfer matrices or quantum circuits in integrable models. A spectral parameter is a continuous variable the R-matrix depends on; most familiar R-matrices depend on one parameter, so the extra parameters here are a notable change.

The construction starts from the star–triangle and star–star relations of the chiral Potts model. The chiral Potts model is a Z_N generalization of the Ising model (Z_N means a clock symmetry with N states). For N>2 its Boltzmann weights depend on two independent variables that live on a curved surface of high genus (a technical way of saying the parameter space is complicated). That property, together with the fact that the quantum Hamiltonians coming from edge-type models include both nearest‑neighbour interactions and onsite potential terms, leads naturally to R-operators that need more than one spectral parameter.

Technically, the author shows how to turn those star–triangle and star–star relations into a Yang–Baxter equation with three spectral parameters. This extends recent work that unifies solvable edge models (like Ising) and vertex models by treating Onsager’s star–triangle relation as a core ingredient rather than an alternative form of the YBE. The paper applies the method to a family of models including the superintegrable von Gehlen–Rittenberg Z_N chain and parafermionic generalizations of the XY and Hubbard constructions. As an explicit example, an interacting R-matrix for the transverse‑field Ising/XY limit with an onsite coupling U is written down; at the critical value U=1 the form simplifies, and the derivative of the R-matrix with respect to a spectral parameter gives a non‑Hermitian but integrable Hamiltonian.

Why this matters: many known integrable quantum chains map to R-matrices that depend on a single spectral parameter. Models with onsite potentials or with the richer algebraic structure of Z_N symmetry have resisted that simple picture. By producing a consistent YBE with more spectral parameters, the work gives a systematic route to include these harder cases in the algebraic framework of integrability. The paper also lays groundwork for parafermions—generalizations of fermions that appear in Z_N models—and for a Fock‑space description of parafermion operators following earlier ideas of Cobanera and Ortiz.