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In this paper, we consider a second order nonlinear ordinary differentialequation of the form  $\ddot{x}+k_1\frac{\dot{x}^2}{x}+(k_2+k_3x)\dot{x}+k_4x^3+k_5x^2+k_6x=0$,where $k_i$'s, $i=1,2,...,6,$ are arbitrary parameters. By using the modifiedPrelle-Singer procedure, we identify five new integrable cases in this equationbesides two known integrable cases, namely (i) $k_2=0, k_3=0$ and (ii) $k_1=0,k_2=0, k_5=0$. Among these five, four equations admit time dependent firstintegrals and the remaining one admits time independent first integral. Fromthe time independent first integral, nonstandard Hamiltonian structure isdeduced thereby proving the Liouville sense of integrability. In the case oftime dependent integrals, we either explicitly integrate the system ortransform to a time-independent case and deduce the underlying Hamiltonianstructure. We also demonstrate that the above second order ordinarydifferential equation is intimately related to the two-dimensionalLotka-Volterra (LV) system. From the integrable parameters of above nonlinearequation and all the known integrable cases of the latter can be deducedthereby.

On certain new integrable second order nonlinear differential equations and their connection with two dimensional Lotka–Volterra system