Can a LFSR has more than one repeat cycle?
There is a very nice recent paper available here by Chang, Ezermann, Ling and Wang, which has addressed this problem for arbitrary alphabet $\mathbb{F}_q$ and arbitrary characteristic polynomials, including those with divisors that are powers of irreducible polynomials.

Let $\Omega(f)$ be the set of sequences satisfying $f$ as their characteristic polynomial.

This paper determines the cycle structure of $\Omega(f)$ fully for an arbitrary $f$ and gives a method to identify a state from each distinct cycle in $\Omega(f)$, a longstanding open problem.