On the interconnections between various analytic approaches in coupled first-order nonlinear differential equations

In the theory of integrable dynamical systems the following quantities, namely Lie point symmetries, λ-symmetries, adjoint symmetries, Darboux polynomials, Jacobi last multiplier and extended Prelle-Singer quantities play a vital role. In this paper, we consider a system of first order nonlinear ordinary differential equations and investigate the following question: Suppose any one of the quantities in the above mentioned list has been given or known whether the rest of the quantities can be derived without going into its own formalism or determining equation? Our analysis shows that there exists a global connection between these quantifiers. The global interconnections have been formulated and demonstrated with suitable examples in the case of systems of two coupled and three coupled first order ODEs, which can then be extended to higher orders.