element $a$ of an $H$-module is called an *$H$-weight vector*, if there exists $\nu\in{\mathfrak{X}}$ such that $h\cdot a=\nu(h)a$ for all $h\in H$. By [@Kostant2 Theorems 6, 7], every $H$-weight occurs in $S({\mathfrak{n}})^N$ with multiplicity at most 1. Furthermore, there exist unique (up to scalars) prime polynomials $\xi_{\beta}\in S({\mathfrak{n}})^N$, $\beta\in{\mathcal{B}}$, such that each $\xi_{\beta}$ is an $H$-weight polynomial of a dominant weight ${\varpi}_{\beta}$ belonging to the ${\mathbb{Z}}$-linear span ${\mathbb{Z}}{\mathcal{B}}$ of ${\mathcal{B}}$. A remarkable fact is that $$\xi_{\beta},~\beta\in{\mathcal{B}},\text{ are algebraically independent generators of }S({\mathfrak{n}})^N,\label{formula:xi_i}$$ so $S({\mathfrak{n}})^N$ and $Z({\mathfrak{n}})$ are polynomial rings. We call $\{\xi_{\beta},~\beta\in{\mathcal{B}}\}$ the set of *canonical generators* of $S({\mathfrak{n}})^N$. Put $m=|{\mathcal{B}}|$. It turns out that the weights ${\varpi}_{\beta}$’s have the following form [@Panov2 Theorem 2.12]. $$\label{formula:weights_of_xi_i}\text{\begin{tabular}{|l|l|}
\hline
$\Phi=A_{n-1}$&${\varpi}_{\beta}=2{\epsilon}_1+\ldots+2{\epsilon}_{i-1}+{\epsilon}_i$ for $\beta={\epsilon}_i-{\epsilon}_{n-i+1}$, $1\leq i\leq m$\\
\hline
$\Phi=B_n$&${\varpi}_{\beta}=\begin{cases}2{\epsilon}_1+\ldots+2{\epsilon}_i&\text{for $\beta={\epsilon}_i-{\epsilon}_{i+1}$ with odd $i<m$},\\
{\epsilon}_1+\ldots+{\epsilon}_i&\text{otherwise}
\end{cases}$\\
\hline
$\Phi=C_n$&${\varpi}_{\beta}=2{\epsilon}_1+\ldots+2{\epsilon}_i$ for $\beta=2{\epsilon}_i$, $1\leq i\leq m$\\
\hline
$\Phi=D_n$&${\varpi}_{\beta}=\begin{cases}2{\epsilon}_1+\ldots+2{\epsilon}_i&\text{for $\beta={\epsilon}_i-{\epsilon}_{i+1}$ with odd $i<m-1$},\\
2{\epsilon}_1+\ldots+2{\epsilon}_{n-1}&\text{for $\beta={\epsilon}_{n-2}-{\epsilon}_{n-1}$ when $n$ is odd},\\
{\epsilon}_1+\ldots+{\epsilon}_{n-1}-{\epsilon}_n&\text{for $\beta={\epsilon}_{n-1}-{\epsilon}_n$ when $n$ is even},\\
{\epsilon}_1+\ldots+{\epsilon}_i&\text{otherwise}
\end{cases}$\\
\hline
\end{tabular}}$$

The following description of $\xi_{\beta}$ for classical root systems, given in [@IgnatyevPenkov1 Subsection 2.1], follows from [@Panov1] (see also [@LipsmanWolf1] and [@Joseph-Faucaunt-Millet1]). Our notation here slightly differs from the one used in [@IgnatyevPenkov1].

i\) $\Phi=A_{n-1}$. Here, for $1\leq i\leq m=[n/2]$, 