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Symmetry analysis of Ramanujan's system of differential equations is performed by representing it as a third-order equation. A new system consisting of a second-order and a first-order equation is derived from Ramanujan's system. The Lie algebra of the new system is equivalent to the algebra of the third-order equation. This forms the basis of our intuition that for a system of first-order odes its infinite-dimensional algebra of symmetries contains a subalgebra which is a representation of the Lie algebra for any system or differential equation which can be obtained from the original system, even though the transformations are not point., Comment: 7 pages

We examine the general element of the class of ordinary differential equations, $yy^{(n+1)}+\alpha y'y^{(n)}=0$, for its Lie point symmetries. We observe that the algebraic properties of this class of equations display an attractive set of patterns, the general member of the class can have three type of Algebra, $(n+1)A_1 \oplus_s\{A_1 \oplus sl(2,R)\}$, $A_1 \oplus sl(2,R)$ or $A_2 \oplus A_1$, for different values of $\alpha$. We look at the singularity properties of these equations for various values of $\alpha$., Comment: 14 pages

We study the symmetry and integrability of a modified Camassa-Holm Equation (MCH), with an arbitrary parameter $k,$ of the form $$u_{t}+k(u-u_{xx})^2u_{x}-u_{xxt}+(u^{2}-{u_{x}}^2)(u_{x}-u_{xxx})=0.$$ By using Lie point symmetries we reduce the order of the above equation and also we obtain interesting novel solutions for the reduced ordinary differential equations. Finally we apply the Painlev\'e Test to the resultant nonlinear ordinary differential equation., Comment: 13 pages

The symmetry analysis of the Cheng Equation is performed. The Cheng Equation is reduced to a first-order equation of either Abel's Equations, the analytic solution of which is given in terms of special functions. Moreover, for a particular symmetry the system is reduced to the Riccati Equation or to the linear nonhomogeneous equation of Euler type. Henceforth, the general solution of the Cheng Equation with the use of the Lie theory is discussed, as also the application of Lie symmetries in a generalized Cheng equation., Comment: 10 pages. Accepted for publication in Quaestiones Mathematicae journal

We consider the master equation of quantum Brownian motion and with the application of the group invariant transformation we show that there exists a surface on which the solution of the master equation is given by an autonomous one-dimensional Schr\"{o}dinger Equation., Comment: 7 pages, to be published in Mathematics (MDPI)