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The local mass is a fundamental quantized information that characterizes theblow-up solution to the Toda system and has a profound relationship with itsunderlying algebraic structure. In \cite{Lin-Yang-Zhong-2020}, it was observedthat the associated Weyl group can be employed to represent this informationfor the $\mathbf{A}_n$, $\mathbf{B}_n$, $\mathbf{C}_n$ and $\mathbf{G}_2$ typeToda systems. The relationship between the local mass of blow-up solution andthe corresponding affine Weyl group is further explored for some affine$\mathbf{B}$ type Toda systems in \cite{Cui-Wei-Yang-Zhang-2022}, where thepossible local masses are explicitly expressed in terms of $8$ types. Thecurrent work presents a comprehensive study of the general affine $\mathbf{A}$and $\mathbf{C}^t$ type Toda systems with arbitrary rank. At each stage of theblow-up process (via scaling), we can employ certain elements (known as "setchains") in the corresponding affine Weyl group to measure the variation oflocal mass. Consequently, we obtain the a priori estimate of the affine$\mathbf{A}$ and $\mathbf{C}^t$ type Toda systems with arbitrary number ofsingularities.

Affine Toda system of $\mathbf{A}$ and $\mathbf{C}^t$ type: compactness and affine Weyl group