We study the symmetry and integrability of a modified Camassa–Holm equation (MCH), with an arbitrary parameter k, of the form

$$\begin{aligned} u_{t}+k(u-u_{xx})^2u_{x}-u_{xxt}+(u^{2}-{u_{x}}^2)(u_{x}-u_{xxx})=0. \end{aligned}$$

The commutator table and adjoint representation of the symmetries are presented to construct one-dimensional optimal system. By using the one-dimensional optimal system, we reduce the order or number of independent variables of the above equation and also we obtain interesting novel solutions for the reduced ordinary differential equations. Finally, we apply the Painlevé test to the resultant nonlinear ordinary differential equation and it is observed that the equation is integrable.

中文翻译：

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具有任意参数的改进的Camassa–Holm方程的对称性和可积性

我们研究具有任意参数k的改进的Camassa-Holm方程（MCH）的对称性和可积性，其形式为

$$ \ begin {aligned} u_ {t} + k（u-u_ {xx}）^ 2u_ {x} -u_ {xxt} +（u ^ {2}-{u_ {x}} ^ 2）（u_ {x} -u_ {xxx}）= 0。\ end {aligned} $$

给出了换向器表和对称性的伴随表示，以构造一维最优系统。通过使用一维最优系统，我们减少了上述方程式的独立变量的阶数或数量，并且对于简化的常微分方程式，我们获得了有趣的新颖解。最后，我们将Painlevé检验应用于所得的非线性常微分方程，并观察到该方程是可积的。