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It has been established that the local mass of blow-up solutions to Todasystems associated with the simple Lie algebras$\mathbf{A}_n,~\mathbf{B}_n,~\mathbf{C}_n$ and $\mathbf{G}_2$ can berepresented by a finite Weyl group. In particular, at each blow-up point, aftera sequence of bubbling steps (via scaling) is performed, the transformation ofthe local mass at each step corresponds to the action of an element in the Weylgroup. In this article, we present the results in the same spirit for theaffine $\mathbf{B}_2^{(1)}$ Toda system with singularities. Compared with theToda system with simple Lie algebras, the computation of local masses is morechallenging due to the infinite number of elements of the {affine Weyl group oftype $\mathbf{B}_{2}^{(1)}$}. In order to give an explicit expression for thelocal mass formula we introduce two free integers and write down all thepossibilities into 8 types. This shows a striking difference to previousresults on Toda systems with simple Lie algebras. The main result of thisarticle seems to provide the first major advance in understanding the relationbetween the blow-up analysis of affine Toda system and the {affine Weyl group}of the associated Lie algebras.

The Blow-up Analysis on $\mathbf{B}_2^{(1)}$ Affine Toda system: Local mass and Affine Weyl group