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