Skew $\sigma$-splittable linear recurrent sequences with maximal period

Let $p$ be a prime number, $R=\mathrm{GR}(q^d,p^d)$ be a Galois ring of cardinality $q^d$ and characteristic $p^d$, where $q = p^r$, $S=\mathrm{GR}(q^{nd},p^d)$ be its extension of degree $n$ and $\sigma$ be a Frobenius automorphism of $S$ over $R$. We study sequences $v$ over $S$ satisfying recursion laws of the form
$$\forall i\in\mathbb{N}_0 \colon v(i+m) = s_{m - 1}\sigma^{k_{m-1}}(v(i+m-1))+\ldots+s_1\sigma^{k_1}(v(i+1)) + s_0\sigma^{k_0}(v(i)),$$
where $s_0,\ldots,s_{m-1}\in S, k_{0},\ldots, k_{m-1}\in \mathbb{N}_{0}$. We say that $v$ is $\sigma$-splittable skew linear recurrent sequence (LRS) over $S$ of order $m$. The period of such LRS is not greater than $(q^{mn}-1)p^{d-1}$. We obtain neccessary and sufficient conditions for $\sigma$-splittable skew LRS to have maximal period. We prove that under some conditions $\sigma$-splittable skew LRS are non-linearized skew LRS. Also we consider linear complexity of such sequences and uniqueness of minimal polynomial over $S$.