The authors carry out a classification of the Lie point symmetry groups associated with the quadratic Liénard type equation \[ \ddot{x}+f(x)\dot{x}^2+g(x) = 0, \eqno{(1)} \] where \(f(x)\) and \(g(x)\) are arbitrary functions of \(x\). The symmetries are divided into two cases, (i) the maximal (eight parameter) symmetry group and (ii) non-maximal (three, two, and one parameter) symmetry groups.
Firstly, the authors consider the linearizable case and find the general form of equation (1) for which it admits eight point symmetry generators. The general form of equation (1) in this case is given by \[ \ddot{x} + f(x) \dot{x}^2 + g_1 e^{-\int f(x) d x} \int e^{\int f(x) dx}dx + g_2 e^{-\int f(x)dx} = 0, \eqno{(2)} \] where \(g_1\) and \(g_2\) are constant parameters.
Secondly, the authors consider the integrable cases of equation (1) with lesser parameter symmetries. The general forms of equations that show three and two parameters symmetry generators, respectively, are given by: \[ \ddot{x} + f(x)\dot{x}^2 + g_1 e^{-\int f(x)dx}\left(\lambda_1 + \int e^{\int f(x) d x} d x\right)^{-3} = 0,\eqno{(3)} \]
\[ \ddot{x}+f(x)\dot{x}^2+g_1e^{-\int f(x) d x} \left(\lambda_1 + \int e^{\int f(x) d x}dx\right)^{1-\lambda_2} = 0,\quad \lambda_2 \not=1,4, \eqno{(4)} \] where \(g_1, \lambda_1\), and \(\lambda_2\) are constants.
The rest of the cases of equation (1) with a Lie point symmetry group corresponds to a one parameter Lie point symmetry group. This case includes the one-dimensional Mathews-Lakshmanan oscillator with \(f(x)=-{\lambda x \over 1+\lambda x^2}\) and \(g(x) = {\omega_0^2 x \over 1+\lambda x^2}\).
This paper also analyzes the underlying equivalence transformations.