  We study the symmetry and integrability of a Generalized Modified
Camassa-Holm Equation (GMCH) of the form
$$u_{t}-u_{xxt}+2nu_{x}(u^2-u_{x}^2)^{n-1}(u-u_{xx})^2+(u^2-u_{x}^2)^{n}(u_{x}-u_{xxx})=0.$$
We observe that for increasing values of $n\in \mathbb{N}$, $\mathbb{N}$
denotes natural number, the above equation gives a family of equations in which
nonlinearity is rapidly increasing as $n$ increases. However, this family has
similar form of symmetries except the values of $n$. Interestingly the
resultant second-order nonlinear ODE which is to be obtained from GMCH equation
has eight dimensional symmetries. Hence the second-order nonlinear ODE is
linearizable. Finally we conclude that the resultant second-order nonlinear
ordinary differential equation which is obtained from the family of GMCH passes
the Painlev\'e Test also it posses the similar form of leading order,
resonances and truncated series solution too.
