On a reduction of the generalized Darboux-Halphen system

The equations for the general Darboux-Halphen system obtained as a reduction of the self-dual Yang-Mills can be transformed to a third-order system which resembles the classical Darboux-Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux-Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.Comment: Some content of this article overlaps with content of another article written previously by two of the authors: arXiv:1606.02910. We have also removed the error, expanding and writing it into a completely new form, several new results having been found. 14 page

Jacobi-Maupertuis metric of Lienard type equations and Jacobi Last Multiplier

We present a construction of the Jacobi-Maupertuis (JM) principle for an equation of the Lienard type, viz \ddot{x} + f(x)x^2 + g(x) = 0 using Jacobi's last multiplier. The JM metric allows us to reformulate the Newtonian equation of motion for a variable mass as a geodesic equation for a Riemannian metric. We illustrate the procedure with examples of Painleve-Gambier XXI, the Jacobi equation and the Henon-Heiles system