Let $H\colon N\to\mathbb{Z}^{r}$ be an injective map such that $H(N)$ is a saturated sublattice of $\mathbb{Z}^{r}$, $p\in\mathbb{P}^{r}(\mathbb{K})$ and $Y\subset\mathbb{P}^{r}$ the closure of the image of the map $\varphi_{H,p}\colon\mathbb{T}\to\mathbb{P}^{r}$. Let $m_{0}\in M$ and $m_{i}=e_{i}^{\vee}\circ H+m_{0}\in M$, $i=1,\dots,r$, and write $p=(p_{0}:\dots:p_{r})$ with $p_{i}\in\mathbb{K}^{\times}$. Let $\Delta=\operatorname{conv}(m_{0},\dots,m_{r})\subset M_{\mathbb{R}}$ and $\vartheta_{v}\colon\Delta\to\mathbb{R}$ the function parameterizing the upper envelope of the extended polytope $\operatorname{conv}\left((m_{0},\log|p_{0}|_{v}),\dots,(m_{r},\log|p_{r}|_{v})\right)\subset M_{\mathbb{R}}\times\mathbb{R}.$ Then $Y$ is integrable and