Let $(u_n)_{n \geq 0}$ be a linear recurrence given by
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n-k} \quad \forall n \geq k ,$$
where $u_0, \ldots, u_{k-1}, a_1, \ldots, a_k \in \mathbb{Z}$. We recall that $(u_n)_{n \geq 0}$ is said to be *degenerate* if its characteristic polynomial $f(x) = X^k - a_1 X^{k-1} - \cdots - a_k$ has two roots $\alpha$ and $\beta$ such that their ratio $\alpha/\beta$ is a root of unity, otherwise $(u_n)_{n \geq 0}$ is said to be *non-degenerate*.

The following theorem tell us that the study of linear recurrence sequences over an algebraic field can effectively be reduced to that of non-degenerate linear recurrence sequences.

**Theorem [1]** Let $(u_n)_{n \geq 0}$ be a linear recurrence of order $k$ over an algebraic number field of degree $d$ over $\mathbb{Q}$. Then there exists an effectively computable constant $M(k,d)$ such that for some $M \leq M(k,d)$ each subsequence $(u_{Mn+\ell})_{n \geq 0}$, with $\ell=0,\ldots,M-1$, is either identically zero or is non-degenerate.

In the light of the previous theorem, usually when dealing with linear recurrence one assumes that these are non-degenerate.

Instead, my question is about *degenerate* linear recurrence with integer coefficient (from now on). In [2] all the second order degenerate linear recurrences are listed. What about higher order? Is some general classification known?

Thank you in advance for any reference.

[1] *G. Everest, A. van der Poorten, and I. Shparlinski, Recurrence sequences, Theorem 1.2*

[2] *P. Ribenboim, My number, my friends, pp. 5--6*

P.S. For who is not practical of linear recurrence sequences, the answer can be equivalently formulated as purely algebraic: Are there some classification of the monic polynomials with integer coefficients $f(X)$ such that there exists two non-zero roots $\alpha$ and $\beta$ of $f(X)$ with $\alpha / \beta$ a root of unity?