Bases for the derivation modules of two-dimensional multi-Coxeter arrangements and universal derivations

Let $\text{A}$ be an irreducible Coxeter arrangement and $\text{bfk}$ be a multiplicity of $\text{A}$. We study the derivation module $D(\text{A},\text{bfk})$. Any two-dimensional irreducible Coxeter arrangement with even number of lines is decomposed into two orbits under the action of the Coxeter group. In this paper, we will {explicitly} construct a basis for $D(\text{A},\text{bfk})$ assuming $\text{bfk}$ is constant on each orbit. Consequently we will determine the exponents of $(\text{A},\text{bfk})$ under this assumption. For this purpose we develop a theory of universal derivations and introduce a map to deal with our exceptional cases.