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In this work, we establish a connection between the extended Prelle-Singerprocedure with other widely used analytical methods to identify integrablesystems in the case of $n^{th}$-order nonlinear ordinary differential equations(ODEs). By synthesizing these methods we bring out the interlink between Liepoint symmetries, contact symmetries, $\lambda$-symmetries, adjoint-symmetries,null forms, Darboux polynomials, integrating factors, Jacobi last multiplierand generalized $\lambda$-symmetries corresponding to the $n^{th}$-order ODEs.We also prove these interlinks with suitable examples. By exploiting theseinterconnections, the characteristic quantities associated with differentmethods can be deduced without solving the associated determining equations.

Interplay of symmetries and other integrability quantifiers in finite dimensional integrable nonlinear dynamical systems