Finding a 3 dimensional manifold that integrates the involutive distribution given by $X=∂/∂x + z∂/∂y$, $Y =∂/∂z + x∂/∂w$ and $Z =∂/∂w - ∂/∂y$ that passes for $(0,0,0,0)$.
I already know that $Z=[X,Y]$ and I dont know if it can help with something.
The approach I am having is: I need to create an exact one form $\omega=df$ such that $\omega(X)=\omega(Y)=\omega(Z)=0$
Let's suppose that $\omega=adx+bdy+cdz+edw=\partial f/\partial xdx+\partial f/\partial ydy+\partial f/\partial zdz+\partial f/\partial wdz$ so I end up with the following system $a+bz=0$ and $c+bx=0$.
As I have $a=\partial f/\partial x, b=\partial f/\partial y, c=\partial f/\partial z$
so $\partial f/\partial x+z\partial f/\partial y=0$ and $\partial f/\partial z+x\partial f/\partial y=0$ it can be solve by solving the system, so, how can I solve this last systems?
Simplifying the notation a bit we have for the vector fields \begin{align} X&=\partial_x+z\,\partial_y\,,&Y=\partial_z+x\,\partial_w \end{align} the commutator \begin{align}[X,Y] &=\partial_w-\partial_y\,. \end{align} A vector field that is orthogonal to $X,Y$ and $[X,Y]$ is \begin{align} U=U^x\,\partial_x+U^y\,\partial_y+U^z\,\partial_z+U^w\,\partial_w\,,\quad\text{ where } \quad U^\mu=\begin{pmatrix}-uz\\u\\-ux\\u\end{pmatrix}\,, \end{align} and $u$ is an arbitrary function of $x,y,z,w\,.$ To specify $u$ further observe that the one-form \begin{align} \boldsymbol{\alpha}&=U^x\,dx+U^y\,dy+U^z\,dz+U^w\,dw\\[2mm] &=u\underbrace{\big(-z\,dx+dy-x\,dz+dw\big)}_{\textstyle=:\,\boldsymbol{\beta}} \end{align} is closed if and only if $$ d\boldsymbol{\alpha}=du\wedge\boldsymbol{\beta}=0\,. $$ This is seen from \begin{align} d\boldsymbol{\alpha}&=du\wedge\boldsymbol{\beta}+u\,d\boldsymbol{\beta} \end{align} and $d\boldsymbol{\beta}=0\,.$ Obviously, the choice $$ u\equiv 1 $$ makes $\boldsymbol{\alpha}$ equal to $\boldsymbol{\beta}$ and therefore closed. It is also easy to see that $\boldsymbol{\beta}=df$ where $$ f(x,y,z,w)=y+w-xz\,. $$ Therefore, for each constant $c$ the solution to $$ f(x,y,z,w)=c $$ is a three-dimensional integral manifold whose normal vector is $U$ and for which $X,Y,[X,Y]$ are tangent vectors (since they are orthogonal to $U$).
Remark. The previous version of this answer used the integrating factor $u=y+w-xz$ and led to a function $f$ such that $\alpha=df\,:$ \begin{align} f(x,y,z,w)=\frac{x^2z^2+y^2+w^2}{2}-xyz-xzw+wy\,. \end{align} But it turns out that this $f$ can be written as $$ f(x,y,z,w)=\frac{(y+w-xz)^2}{2}\,. $$ So the integral manifolds are the same, as they should.