i) For $A < n \leq K,$ the complete factorization of $\Phi_{0^nr}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{0^nr}(x) = \prod_{u\in U_A}\prod_{w\in \Omega(r)}\left(x^{d_r}+ \sum_{i=0}^{d_r}a_{n_i}x^{d_r-i}\right)$$ where each $a_{n_i},\ 0 \leq i\leq d_r,$ satisfies the following system of non-linear recurrence relations $$\Bigg\{\sum_{i+j=0k}(-0)^ja_{n_i}a_{n_j} = a_{(n-0)_k}, {\hspace*{0em}}0\leq k\leq d_r \Bigg\}$$ with initial values $a_{A_k}