Let us consider the Korteweg-de Vries (KdV) equation. Upon applying the Miura transformation to this equation, it is reformulated as follows: \begin{align} \mathcal{E}_1=\left(\frac{1}{2}u+\partial_x\right)\left(\dot{u}+\frac{3}{16}u^2u'-\frac{a}{2}u'''\right)=0. \end{align} The equation \begin{align} \dot{u}+\frac{3}{16}u^2u'-\frac{a}{2}u'''=0, \end{align} is recognized as the modified Korteweg-de Vries (MKdV) equation. A solution to the MKdV equation will solve $\mathcal{E}_1$, but the converse is not necessarily true; a solution to $\mathcal{E}_1$ may not satisfy the MKdV equation. This raises the question: Is there a method to map a solution from $\mathcal{E}_1$ to the MKdV equation?
This inquiry leads to a broader problem. Considering a specific partial differential equation $\mathrm{PDE}$, which adheres to the system \begin{align} \mathcal{E}_2=\mathcal{D}\left(\mathrm{PDE}\right)=0, \end{align} one might ask whether it is possible to map solutions of $\mathcal{E}_2$ to those satisfying $\mathrm{PDE}=0$.