  In this work, we establish a connection between the extended Prelle-Singer
procedure (Chandrasekar \textit{et al.} Proc. R. Soc. A 2005) with five other
analytical methods which are widely used to identify integrable systems in the
contemporary literature, especially for second order nonlinear ordinary
differential equations (ODEs). By synthesizing these methods we bring out the
interplay between Lie point symmetries, $\lambda$-symmetries, adjoint
symmetries, null-forms, Darboux polynomials, integrating factors and Jacobi
last multiplier in identifying the integrable systems described by second order
ODEs. We also give new perspectives to the extended Prelle-Singer procedure
developed by us. We illustrate these subtle connections with modified Emden
equation as a suitable example.
