If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D Monge-Ampere equation. so if we extend our Jacobi PDE system to 3-forms instead of 2-forms and assume conservation law $\theta \in \Omega^2(M)$ such that 3-form $d \theta$ is non-degenerate 3-form, then the new Jacobi PDE system can be written locally as the generalized 3D Symplectic Monge-Ampere equation arising from 3-forms ?

PS: Firstly, I try to explain more the question and will try to solve it for 2D symplectic monge ampere equation and Jacobi plane and after solve the question for 3D, but I am still looking for an **affirmative answer**.

Let me explain the Jacobi PDE system view of 2D symplectic Monge Ampere equation .In fact by using **Darboux Theorem** we can give a nice answer to this question in 2D case . So here, First I prove the following proposition:

But before I try to define Jacobi PDE system for readers as follows

**Definition** : We call following system as Jacobi PDE system

$a_1+b_1\frac{\partial h_1}{\partial x_1}-c_1\frac{\partial h_1}{\partial x_2}-d_1\frac{\partial h_2}{\partial x_2}+e_1\frac{\partial h_2}{\partial x_1}+f_1\left (\frac{\partial h_1}{\partial x_1}\frac{\partial h_2}{\partial x_2}-\frac{\partial h_1}{\partial x_2}\frac{\partial h_2}{\partial x_1} \right )=0$ $a_2+b_2\frac{\partial h_1}{\partial x_1}-c_2\frac{\partial h_1}{\partial x_2}-d_2\frac{\partial h_2}{\partial x_2}+e_2\frac{\partial h_2}{\partial x_1}+f_2\left (\frac{\partial h_1}{\partial x_1}\frac{\partial h_2}{\partial x_2}-\frac{\partial h_1}{\partial x_2}\frac{\partial h_2}{\partial x_1} \right )=0$

and we can correspond each equation in Jacobi PDE system as symplectic 2-form $\omega$. So we correspond a Jacobi PDE system to two 2-from $\omega_1, \omega_2$ ,also it is clear that the combination of $\omega_1$ and $ \omega_2$ are also Jacobi PDE equation, so we can correspond each Jacobi PDE system as $\prod=<\omega_1, \omega_2>$.

**Proposition:** Let $\prod=<\omega_1,\omega_2>$ be a Jacobi PDE system with a conswerwation law $\theta \in \Omega^1(M)$ , such that $d\theta=a\omega_1+b\omega_2$ is non-degenerate 2-form , then locally the Jacobi PDE system can be written as Monge-Ampere equation.

The reason is: In fact the symplectic Monge-Ampere equation have the following form

$\hat{a}+b\frac{\partial^2\varphi}{\partial x_1^2}-d\frac{\partial^2\varphi}{\partial x_2^2}-c\frac{\partial^2\varphi}{\partial x_2\partial x_1}+e\frac{\partial^2\varphi}{\partial x_1\partial x_2}+\check{a}\frac{\partial\varphi}{\partial x_1}+\tilde{a}\frac{\partial\varphi}{\partial x_2}+$

$f(\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_2^2}-\frac{\partial^2\varphi}{\partial x_2\partial x_1}\frac{\partial^2\varphi}{\partial x_1\partial x_2})=0$

where the coefficients are smooth functions of $x$ and $\frac{\partial\varphi}{\partial x}$ .

By the following trick, we reduce the symplectic Monge-Ampere equation into a Jacobi PDE system and vise versa. In fact by substitution $h_1=\frac{\partial\varphi}{\partial x_1}$, and $h_2=\frac{\partial\varphi}{\partial x_2}$ and taking following compatibility condition

$\frac{\partial h_2}{\partial x_1}=\frac{\partial h_1}{\partial x_2}$ we get the following Jacobi PDE system:

$\frac{\partial h_2}{\partial x_1}-\frac{\partial h_1}{\partial x_2}=0$

$a+b\frac{\partial h_1}{\partial x_1}-c\frac{\partial h_1}{\partial x_2}-d\frac{\partial h_2}{\partial x_2}+e\frac{\partial h_2}{\partial x_1}+f(\frac{\partial h_1}{\partial x_1}\frac{\partial h_2}{\partial x_2}-\frac{\partial h_1}{\partial x_2}\frac{\partial h_2}{\partial x_1})=0$

where $a=\hat{a}+\breve{a}h_1+\tilde{a}h_2$ .

Therefore the corresponding 2-forms of this system are:

$\omega_1=dx_1\wedge du_1+dx_2\wedge du_2$

$\omega_2=a(x,u)dx_1\wedge dx_2+ b(x,u)du_1\wedge dx_2+c(x,u)du_1\wedge dx_1+$

$d(x,u)du_2\wedge dx_1 +e(x,u)du_2\wedge dx_2+f(x,u)du_1\wedge du_2$

So the non-degenerate 2-form $d\theta$ , determines a symplectic structure on $M$ .Moreover by applying Darboux theorem, locally there exists a canonical coordinate system for $d\theta $, say $(x_1,x_2,u_1,u_2)$, such that : $d\theta=dx_1\wedge du_1+dx_2\wedge du_2$.

Now, let $\omega'$ be a 2-form, such that $<\omega', d\theta>$ is a local basis for $\prod$ . Then $\omega'$ has the same form as $\omega_2$ as above. Therefore the Jacobi PDE system $d\theta$ and $\omega'$ can be written as the 2D symplectic Monge-Ampere equation as above.

So if we continue this method for 3D symplectic monge ampere equation, we first define 3D Jacobi PDE system(this is my definition for compatibility of 3D Monge-Ampere equation and generalized Jacobi PDE system, please check it yourself again)

**Definition** (**3D Jacobi PDE system**): We call following system(4 PDE equations) as a 3D Jacobi PDE system ($k=1,2,3,4$)
$a^k(x,h(x))det\pmatrix{\frac{\partial u_1}{\partial x_1} & \frac{\partial u_1}{\partial x_2}&\frac{\partial u_1}{\partial x_3}\cr \frac{\partial u_2}{\partial x_1} & \frac{\partial u_2}{\partial x_2}&\frac{\partial u_2}{\partial x_3}\cr \frac{\partial u_3}{\partial x_1}&\frac{\partial u_3}{\partial x_2}&\frac{\partial u_3}{\partial x_3}\cr} +\sum_{i,j}b^k_{ij}(x,h(x))det \pmatrix{\frac{\partial u_i}{\partial x_1} & \frac{\partial u_i}{\partial x_2}\cr \frac{\partial u_j}{\partial x_1} & \frac{\partial u_j}{\partial x_2}\cr} -$

$\sum_{ij}k^k_{ij}(x,h(x))\pmatrix{\frac{\partial u_i}{\partial x_1} & \frac{\partial u_i}{\partial x_3}\cr \frac{\partial u_j}{\partial x_1} & \frac{\partial u_j}{\partial x_3}\cr}+\sum_{ij}c^k_{ij}(x,h(x))\pmatrix{\frac{\partial u_i}{\partial x_1} & \frac{\partial u_i}{\partial x_3}\cr \frac{\partial u_j}{\partial x_2} & \frac{\partial u_j}{\partial x_3}\cr}+$

$\sum_i m_i^k (x,h(x))\frac{\partial u_i}{\partial x_1}-\sum_i n_i^k (x,h(x))\frac{\partial u_i}{\partial x_2}+\sum_i l_i^k (x,h(x))\frac{\partial u_i}{\partial x_3}+e^k(x,h)=0$

, my computation show that by reducing 3D symplectic monge ampere equation to Jacobi PDE system we have four 3-forms. So if we assume $\prod=<\omega_1,\omega_2,\omega_3,\omega_4>$ be a 3D Jacobi PDE system with a conserwation law $\theta \in \Omega^2(M)$ such that $d\theta=a\omega_1+b\omega_2+c\omega_3+d\omega_4$ be a non-degenerate 3-form. In fact $\omega_1,\omega_2,\omega_3,\omega_4$ have the following 3-forms

$\omega_1=dx_1\wedge dx_2 \wedge du_1+dx_3\wedge dx_2\wedge du_3$

$\omega_2=dx_3\wedge dx_2 \wedge du_2+dx_3\wedge dx_1\wedge du_1$

$\omega_3=dx_1\wedge dx_2 \wedge du_2-dx_3\wedge dx_1\wedge du_3$

$\omega_4=du_1\wedge du_2 \wedge du_3+\sum_{ij}\sum_{k=1}^n a_{ij}^k du_i\wedge du_j\wedge dx_k+$

$\sum_i\sum_{j,k}b_{jk}^i du_i\wedge dx_j \wedge dx_k+dx_1\wedge dx_2 \wedge dx_3$

In fact, $\omega_1,\omega_2, \omega_3$ come of the following compatibility conditions:

$\frac{\partial h_1}{\partial x_2}=\frac{\partial h_2}{\partial x_1}$

$\frac{\partial h_2}{\partial x_3}=\frac{\partial h_3}{\partial x_2}$

$\frac{\partial h_1}{\partial x_3}=\frac{\partial h_3}{\partial x_1}$

So we obtain , the Jacobi PDE system $\prod=<\omega_1,\omega_2,\omega_3,\omega_4>$ can be written as following system:

3-D symplectic monge Ampere equation = $A=a(x,h(x))[\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2\varphi}{\partial x_3^2}+\frac{\partial^2\varphi}{\partial x_1 \partial x_2}\frac{\partial^2\varphi}{\partial x_2 \partial x_3}\frac{\partial^2\varphi}{\partial x_3 \partial x_1}+\frac{\partial^2\varphi}{\partial x_2 \partial x_1}\frac{\partial^2\varphi}{\partial x_3 \partial x_2}\frac{\partial^2\varphi}{\partial x_1 \partial x_3}]-$

$(\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_2 \partial x_3}\frac{\partial^2\varphi}{\partial x_3 \partial x_2}+\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2\varphi}{\partial x_1 \partial x_3}\frac{\partial^2\varphi}{\partial x_3 \partial x_1}+\frac{\partial^2\varphi}{\partial x_3^2}\frac{\partial^2\varphi}{\partial x_1 \partial x_2}\frac{\partial^2\varphi}{\partial x_2 \partial x_1})+$

$b(x,h(x))[\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_2^2}-\frac{\partial^2\varphi}{\partial x_1 \partial x_2}\frac{\partial^2\varphi}{\partial x_2 \partial x_1}]+c(x,h(x))[\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_3^2}-\frac{\partial^2\varphi}{\partial x_1 \partial x_3}\frac{\partial^2\varphi}{\partial x_3 \partial x_1}]+$

$d(x,h(x))[\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2\varphi}{\partial x_3^2}-\frac{\partial^2\varphi}{\partial x_2 \partial x_3}\frac{\partial^2\varphi}{\partial x_3 \partial x_2}]+e(x,h(x))[\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_2 \partial x_3}-\frac{\partial^2\varphi}{\partial x_1 \partial x_2}\frac{\partial^2\varphi}{\partial x_1 \partial x_3}]-$

$f(x,h(x))[\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2\varphi}{\partial x_1 \partial x_3}-\frac{\partial^2\varphi}{\partial x_1 \partial x_2}\frac{\partial^2\varphi}{\partial x_2 \partial x_3}]+g(x,h(x))[\frac{\partial^2\varphi}{\partial x_3^2}\frac{\partial^2\varphi}{\partial x_1 \partial x_2}-\frac{\partial^2\varphi}{\partial x_1 \partial x_3}\frac{\partial^2\varphi}{\partial x_2 \partial x_3}]+$

$h'(x,h(x))\frac{\partial^2\varphi}{\partial x_1^2}-i(x,h(x))\frac{\partial^2\varphi}{\partial x_2^2}+j(x,h(x))\frac{\partial^2\varphi}{\partial x_3^2}+k(x,h(x))\frac{\partial^2\varphi}{\partial x_1 \partial x_2}-$

$l(x,h(x))\frac{\partial^2\varphi}{\partial x_2 \partial x_3}+m(x,h(x))\frac{\partial^2\varphi}{\partial x_1 \partial x_3}$

$n(x,h(x))\frac{\partial \varphi}{\partial x_1}+o(x,h(x))\frac{\partial \varphi}{\partial x_2}+p(x,h(x))\frac{\partial \varphi}{\partial x_3}+q(x,h(x))=0$

&

$\frac{\partial^2\varphi}{\partial x_1\partial x_2}=\frac{\partial^2\varphi}{\partial x_2\partial x_1}$

$\frac{\partial^2\varphi}{\partial x_1\partial x_3}=\frac{\partial^2\varphi}{\partial x_3\partial x_1}$

$\frac{\partial^2\varphi}{\partial x_2\partial x_3}=\frac{\partial^2\varphi}{\partial x_3\partial x_2}$

But automatically we have last three equations, because $h_i\in C^1$ so $\varphi\in C^2$ and by substituting these equations in first equation $A$ we get the generic 3D Monge-Ampere equation.

fourPDE of the given type (i.e., linear in the minors of the Jacobian matrix with coefficients that depend on the variables $x_i$ and the unknowns $u_i$ (which you are also confusing with the $h_i$, but that's minor)). I think that the reason you want to do this is that you get a system of four equations when you convert a symplectic Monge-Ampère equation (which is second order) to a first order PDE system. However, the system of four first order equations that you get from this conversion isveryspecial. (cont.) $\endgroup$8more comments