The notions of partial Lagrangians, partial Noether operators and partial Euler-Lagrange equations are used in the construction of first integrals for ordinary differential equations (ODEs) that need not be derivable from variational principles. We obtain a Noetherlike theorem that provides the first integral by means of a formula which has the same structure as the Noether integral. However, the invariance condition for the determination of the partial Noether operators is different as we have a partial Lagrangian and as a result partial Euler-Lagrange equations. In order to investigate the effectiveness of the partial Lagrangian approach, some models such as the oscillator systems both linear and nonlinear, Emden and Ermakov-pinnery equations and the Hamiltonian system with two degrees of freedom are considered in this work. We study a general linear system of two second-order ODEs with variable coefficients. Note that, a Lagrangian exists for the special case only but, in general, the system under consideration does not have a standard Lagrangian. However, partial Lagrangians do exist for all such equations in the absence of Lagrangians. Firstly, we classify all the Noether and partial Noether operators for the case when the system admits a standard Lagrangian. We show that the first integrals that result due to the partial Noether approach is the same as for the Noether approach. First integrals are then constructed by the partial Noether approach for the general case when there is in general no Lagrangian for the system of two second-order ODEs with variable coefficients. We give an easy way of constructing first integrals for such systems by utilization of a partial Noether’s theorem with the help of partial Noether operators associated with a partial Lagrangian. Furthermore, we classify all the potential functions for which we construct first integrals for a system with two degrees of freedom. Moreover, the comparison of Lagrangian and partial Lagrangian approaches for the two degrees of freedom Lagrangian system is also given. In addition, we extend the idea of a partial Lagrangian for the perturbed ordinary differential equations. Several examples are constructed to illustrate the definition of a partial Lagrangian in the approximate situation. An approximate Noether-like theorem which gives the approximate first integrals for the perturbed ordinary differential equations without regard to a Lagrangian is deduced. We study the approximate partial Noether operators for a system of two coupled nonlinear oscillators and the approximate first integrals are obtained for both resonant and non-resonant cases. Finally, we construct the approximate first integrals for a system of two coupled van der Pol oscillators with linear diffusive coupling. Since the system mentioned above does not satisfy a standard Lagrangian, the approximate first integrals are still constructed by invoking an approximate Noether-like theorem with the help of approximate partial Noether operators. This approach can give rise to further studies in the construction of approximate first integrals for perturbed equations without a variational principle.

This work is devoted to investigating the algebraic and analytic integrability of first order polynomial partial differential equations via an understanding of the well-developed area of local and global integrability of polynomial vector fields. In the view of characteristics method, the search of first integrals of the first order partial differential equations P(x,y,z)∂z(x,y) ∂x +Q(x,y,z)∂z(x,y) ∂y = R(x,y,z), (1) is equivalent to the search of first integrals of the system of the ordinary differential equations dx/dt= P(x,y,z), dy/dt= Q(x,y,z), dz/dt= R(x,y,z). (2) The trajectories of (2) will be found by representing these trajectories as the intersection of level surfaces of first integrals of (1). We would like to investigate the integrability of the partial differential equation (1) around a singularity. This is a case where understanding of ordinary differential equations will help understanding of partial differential equations. Clearly, first integrals of the partial differential equation (1), are first integrals of the ordinary differential equations (2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2 are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary function of φ1and φ2. We choose for our investigation a system with quadratic nonlinearities and such that the axes planes are invariant for the characteristics: this gives three dimensional Lotka– Volterra systems x' =dx/dt= P = x(λ +ax+by+cz), y' =dy/dt= Q = y(µ +dx+ey+ fz), z' =dz/dt= R = z(ν +gx+hy+kz), where λ,µ,ν 6= 0. v Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions for both integrability and linearizability for three dimensional Lotka-Volterra systems for (1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable. Also, more general three dimensional system have been investigated and necessary and sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems. These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder. We also give a generalization of Singer’s theorem about the existence of Liouvillian first integrals in codimension 1 foliations in Cnas well as to three dimensional vector fields. Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate from the periodic orbits of a centre of a cubic homogeneous polynomial system using the averaging theory of first order.