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Let $\\mathcal S(n)$ be the category of such systems where the operator $T$ acts with nilpotency index at most $n$. We determine the dimension types $(\\dim U_1, \\dim U_2/U_1, \\dim V/U_2)$ of indecomposable systems in $\\mathcal S(n)$ for $n\\leq 4$. It turns out that in the case where $n=4$ there are infinitely many such triples $(x,y,z)$, they all lie in the cylinder given by $|x-y|,|y-z|,|z-x|\\leq 4$. 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