operator $\what\R$ for these flows consists of a hierarchy of higher symmetries $\X^{(n)}=\what P^{(n)}\partial_v$, $n=1,2,\ldots$, that are admitted by every equation $v_t= \what P^{(n)}$, $n=1,2,\ldots$, in the hierarchy.

The first flow in the resulting hierarchy is given by a linear combination of the root flow and the first flow in the mCH hierarchy : $\hat P^{(1)} = \wtil P^{(1)} - P^{(0)} = \what\R(-v_x)  = (\R-1)(-u_x)$ where $$\R(-u_x) = -u_{xxx} - \tfrac{1}{2}((u^2-u_x^2)v)_x .$$ This yields $$\label{mch-variant-flow}
\hat P^{(1)} = \wtil P^{(1)} - P^{(0)} = (\R-1)(-u_x) = v_x - \tfrac{1}{2}((u^2-u_x^2)v)_x .$$ Since all flows in that hierarchy have a bi-Hamiltonian structure given by the pair of mCH Hamiltonian operators , the flow is also bi-Hamiltonian. The corresponding bi-Hamiltonian equation $v_t=\hat P^{(1)}$ is a slight generalization of the mCH equation : $$\label{mch-variant-eqn}
v_t - v_x + \tfrac{1}{2} ((u^2-u_x^2)v)_x =0 .$$ In particular, a Galilean transformation $x\rightarrow \tilde x=x+t$, $t\rightarrow\tilde t=t$ applied to the mCH equation yields the generalized equation .

Lax pair
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The shifted recursion operator , which generates linear combinations of flows in the mCH hierarchy , is closely connected to a Lax pair for the mCH equation .

To explain the connection, we first consider the mKdV shifted recursion operator 