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

Let $A$ be an irreducible Coxeter arrangement and $bfk$ be a multiplicity of $A$. We study the derivation module $D(A, bfk)$. 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 $D(A, bfk)$ assuming $bfk$ is constant on each orbit. Consequently we will determine the exponents of $(A, bfk)$ under this assumption. For this purpose we develop a theory of universal derivations and introduce a map to deal with our exceptional cases.