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A mapping of the same type as the [[Darboux transformation|Darboux transformation]]: it connects the solutions and the coefficients of equations

\begin{equation*} \psi _ { x y } + u ( x , y ) \psi = 0 \end{equation*}

so that if $\varphi$ and $\psi$ are different solutions of it, then the solution of the twin equation with $\psi \rightarrow \psi [ 1 ]$, $u ( x , y ) \rightarrow u [ 1 ] ( x , y )$ may be constructed as the solution of the system

\begin{equation*} ( \psi [ 1 ] \varphi ) _ { x } = - \varphi ^ { 2 } ( \psi \varphi ^ { - 1 } ) _ { x }, \end{equation*}

\begin{equation*} ( \psi [ 1 ] \varphi ) _ y = \varphi ^ { 2 } ( \psi \varphi ^ { - 1 } ) _ y. \end{equation*}

The transformed coefficient (a potential in mathematical physics) is given by

\begin{equation*} u [ 1 ] = u - 2 ( \operatorname { log } \varphi ) _ { x y } = - u + \frac { \varphi _ { x } \varphi_y } { \varphi ^ { 2 } }; \end{equation*}

in other words,

\begin{equation*} \psi [ 1 ] = \psi - \frac { \varphi \Omega ( \varphi , \psi ) } { \Omega ( \varphi , \varphi ) }, \end{equation*}

where $\Omega$ is the integral of the exact differential form

\begin{equation*} d \Omega = \varphi \psi _ { x } d x + \psi \varphi_y d y. \end{equation*}

The important feature defining it is that the transform is parametrized by a pair of solutions of the equation and that the transform vanishes if the solutions coincide.

Clearly, the Moutard equation can be transformed to a $2$-dimensional Schrödinger equation (cf. also [[Schrödinger equation|Schrödinger equation]]), and can be studied in connection with the central problems of classical [[Differential geometry|differential geometry]]. In the theory of solitons (cf. [[Soliton|Soliton]]) it enters via Lax pairs for non-linear equations as the Nizhik–Veselov–Novikov equations [[#References|[a1]]], [[#References|[a2]]].

In [[#References|[a3]]] the Moutard transformation appears in the context of Painlevé analysis.

There is a generalization of Moutard transformations to higher dimensions [[#References|[a4]]]; a proof of the (local) completeness can be found in [[#References|[a5]]].

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  C. Athorne,   J.J.C. Nimmo,   "On the Moutard transformation for integrable partial differential equations"  ''Inverse Problems'' , '''7'''  (1991)  pp. 809–826 {{ZBL|0737.35091}}</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top">  V.B. Matveev,   M.A. Salle,   "Darboux transformations and solitons" , Springer  (1991)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  P.G. Estevez,   S. Leble,   "A wave equation in $2 n + 1$: Painlevé analysis and solutions"  ''Inverse Problems'' , '''11'''  (1995)  pp. 925–937</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C. Athorne,   "On the characterization of Moutard transformations"  ''Inverse Problems'' , '''9'''  (1993)  pp. 217–232</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  E. Gahzha,   "On completeness of the Moutard transformations"  ''solv-int@xyz.lanl.gov'' , '''9606001'''  (1996)</td></tr></table>