  We unearth the interconnection between various analytical methods which are
widely used in the current literature to identify integrable nonlinear
dynamical systems described by third-order nonlinear ordinary differentiable
equations (ODEs). We establish an important interconnection between extended
Prelle-Singer procedure and {\lambda}-symmetries approach applicable to
third-order ODEs to bring out the various linkages associated with these
different techniques. By establishing this interconnection we demonstrate that
given any one of the quantities as a starting point in the family consisting of
Jacobi last multipliers, Darboux polynomials, Lie point symmetries,
adjoint-symmetries, {\lambda}-symmetries, integrating factors and null forms
one can derive the rest of the quantities in this family in a straightforward
and unambiguous manner. We also illustrate our findings with three specific
examples.
