  We show that for a given Nakayama algebra $\Theta$, there exist countably
many cyclic Nakayama algebras $\Lambda_i$, where $i \in \mathbb{N}$, such that
the syzygy filtered algebra of $\Lambda_i$ is isomorphic to $\Theta$ and we
describe those algebras $\Lambda_i$. We show, among these algebras, there
exists a unique algebra $\Lambda$ where the defects, representing the number of
indecomposable injective but not projective modules, remain invariant for both
$\Theta$ and $\Lambda$. As an application, we achieve the classification of
cyclic Nakayama algebras that are minimal Auslander-Gorenstein and dominant
Auslander-regular algebras of global dimension three. Specifically, by using
the Auslander-Iyama correspondence, we obtain cluster-tilting objects for
certain Nakayama algebras. Additionally, we introduce cosyzygy filtered
algebras and show that it is dual of syzygy filtered algebra.
