I found in this book, on page 6 that the equation on vertices of icosahedron inscribed in sphere considered as $\mathbb{CP}^1$ by means of stereographic projection is $xy(x^{10}+14x^5y^5-y^{10})=0$.
It seems that for other polyhedra there are similar equation, so how can I simultaneously understand why all of them are integral, and can I calculate them more easily?