Conformal <i>r</i>-matrix-Nijenhuis structures, symplectic-Nijenhuis structures, and ON-structures

In this paper, we first study infinitesimal deformations of a Lie conformal algebra and with module (called an [Formula: see text] pair), which lead to the notions Nijenhuis operator on structure pair, respectively. Then, by adding compatibility conditions between structures text]-operators, introduce notion text]-structure pair show that gives rise hierarchy pairwise compatible text]-operators. particular, text]-operators can be characterized operators algebras. Finally, r-matrix-Nijenhuis symplectic-Nijenhuis algebras their relations.