If we have a [Jacobi PDE system](https://mathoverflow.net/questions/112173/under-which-conditions-jacobi-pde-system-can-be-represented-to-symplectic-monge-a) 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.



conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables
I think that you should be careful to define your terms, but let me guess: A *Jacobi PDE system* for three unknown functions $h\_1,h\_2,h\_3$ of three independent variables $x\_1,x\_2,x\_3$ is a set of three partial differential equations, each of which can be written as a linear combination of the minors (of any rank, including $0$ and $3$) of the Jacobian matrix $J\_x(h) = \frac{\partial h}{\partial x}$ with coefficients that are explicit functions of the $6$ variables $x\_1,x\_2,x\_3,h\_1,h\_2,h\_3$. In other words, on $M=\mathbb{R}^6$ (or some open subset) with coordinates $x\_1,x\_2,x\_3,h\_1,h\_2,h\_3$, you have specified three $3$-forms $\Upsilon\_i$ ($i=1,2,3$), and a graph 
$$\Gamma\_u = \bigl\lbrace\bigl(x,u(x)\bigr)\ \mid\ x\in D\subseteq\mathbb{R}^3\bigr\rbrace
$$ solves your system with $h=u(x)$ for some function $u:D\to\mathbb{R}^3$ if and only if the pullback of each of the $\Upsilon\_i$ to $\Gamma\_u$ vanishes. To get a decent theory, you'll also need to impose some kind of nondegeneracy condition, such as the condition that, for a 'generic' pair of tangent vectors $v\_1,v\_2\in T\_pM$, the $1$-forms $\theta\_i(v) = \Upsilon\_i(v\_1,v\_2,v)$ are linearly independent at $p$.


Now, it turns out that such a system is *never* equivalent to a so-called *symplectic Monge-Ampère system*, which is locally described on a $6$-manifold $M$ by a choice of a symplectic $2$-form $\Omega$ on $M$ and a nonzero $3$-form $\Upsilon$ that is $\Omega$-primitive, i.e., such that $\Omega\wedge\Upsilon=0$. (In this case, the $3$-manifolds you want to study are the $\Omega$-Lagrangian submanifolds $L\subset M$ to which $\Upsilon$ pulls back to be the zero $3$-form. 


The reason is that for Jacobi PDE systems as above (that satisfy the nondegeneracy condition), the general solution depends locally on *three* arbitrary functions of $2$-variables while, for the symplectic Monge-Ampère systems as defined above, the general solution depends locally on *two* arbitrary functions of $2$-variables.


On the other hand, you could consider a generalization of 'symplectic Monge-Ampère system' in which you have a differential ideal $\mathcal{I}$ on a $6$-manifold that is generated by a nondegenerate $2$-form $\Omega$ and a nonvanishing $\Omega$-primitive $3$-form $\Upsilon$. The condition that they generate a *differential* ideal is just that there exist a $1$-form $\alpha$ and a function $a$ such that 
$$
d\Omega = \alpha\wedge\Omega + a\ \Upsilon.
$$
(If $a\equiv0$, then one finds that $d\alpha=0$, so that, at least locally, one can scale $\Omega$ to make it be closed, and you are back in the symplectic Monge-Ampère case. However, if $a\not=0$, you cannot do this.) In this generalized situation, the general solution will depend on two arbitrary functions of $2$ variables, just as in the symplectic Monge-Ampère case. It would now make sense to think of conservation laws as closed $3$-forms in $\mathcal{I}$, and there are interesting question to ask there.


