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We consider an affine Kac-Moody algebra $\mathfrak{g}$ with Cartan subalgebra$\mathfrak{h}$. Let $\mathfrak{g}[u]$ be a winding subalgebra of$\mathfrak{g}$. Given $\Lambda$ (resp. $\lambda$) in the set $P_+$ (resp.$\dot{P}_+$) of dominant integral weights of $\mathfrak{g}$ (resp.$\mathfrak{g}[u]$), we denote by $L(\Lambda)$ (resp. $\dot{L}(\lambda)$) theintegrable highest weight $\mathfrak{g}$-module (resp.$\mathfrak{g}[u]$-module) with highest weight $\Lambda$ (resp. $\lambda$). The$\mathfrak{g}$-module $L(\Lambda)$ decomposes as a direct sum of weight spaces$L(\Lambda)_\mu$ ($\mu \in \mathfrak{h}^*$). It also decomposes as a direct sumof $\mathfrak{g}[u]$-modules $\dot{L}(\lambda)$. We are interested in thesupports of these decompositions $\Gamma(\mathfrak{g},\mathfrak{h})$ and$\Gamma(\mathfrak{g},\mathfrak{g}[u])$. We show that they are semigroups. Let$P(\Lambda)$ (resp. $P_{\mathfrak{g},u}(\Lambda)$) be the set of all $\lambda\in \mathfrak{h}^*$ (resp. $\dot{P}_+$) such that $L(\Lambda)_\lambda \ne 0$(resp. $\dot{L}(\lambda) \subset L(\Lambda)$). Let $\delta$ be the basisimaginary root of $\mathfrak{g}$. For each $\lambda \inP(\Lambda)+\mathbb{C}\delta$ (resp.$P_{\mathfrak{g},u}(\Lambda)+\mathbb{C}\delta$), we set $b_{\Lambda,\lambda}$(resp. $b_{\Lambda,\lambda,u}$) the complex number $b$ such that$\lambda+b\delta \in P(A)$ (resp. $P_{\mathfrak{g},u}(\Lambda)$) and$\lambda+(b+n)\delta \not\in P(\Lambda)$ (resp. $P_{\mathfrak{g},u}(\Lambda)$)for any $n\in \mathbb{Z}_{>0}$. For the cases $A^{(1)}_1$ and $A^{(2)}_2$, wedetermine explicitly the number $b_{\Lambda,\lambda}$ and a set$\mathcal{A}_u(\Lambda)$ of $\lambda$ satisfying$b_{\Lambda,\lambda,u}=b_{\Lambda,\lambda}$. This help us realize the relationbetween $\Gamma(\mathfrak{g},\mathfrak{g}[u])$ and its satured setting.

Branching problem on winding subalgebras of affine Kac-Moody algebras A^{(1)}_1 and A^{(2)}_2