Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O}N$-structures

In this paper, we first study infinitesimal deformations of a Lie conformal algebra and a Lie conformal algebra with a module (called an $\mathsf{L}\mathsf{C}\mathsf{M}\mathsf{o}\mathsf{d}$ pair), which lead to the notions of Nijenhuis operator on the Lie conformal algebra and Nijenhuis structure on the $\mathsf{L}\mathsf{C}\mathsf{M}\mathsf{o}\mathsf{d}$ pair, respectively. Then by adding compatibility conditions between Nijenhuis structures and $\mathcal{O}$-operators, we introduce the notion of an $$\mathcal{O}$ N$-structureonan$$\mathsf{L}\mathsf{C}\mathsf{M}\mathsf{o}\mathsf{d}$$pairandshowthatan$$\mathcal{O}$ N$-structuregivesrisetoahierarchyofpairwisecompatible$$\mathcal{O}$$-operators\mathrm{.}Inparticular,weshowthatcompatible$$\mathcal{O}$$-operators on a Lie conformal algebra can be characterized by Nijenhuis operators on Lie conformal algebras. Finally, we introduce the notions of conformal $r$-matrix-Nijenhuis structure and symplectic-Nijenhuis structure on the Lie conformal algebra and study their relations.