Describe the maximal integral submanifolds of a distribution spanned by the following vector fields My question: 

Describe the maximal integral submanifolds  of a distribution spanned by the vector fields $$X=y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y}, \ \ \ \ \ \ \ \ \ Y=z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}$$

My Attempt:

I am not sure how to go about trying to solve this problem.

Some background:

A submanifold $(M,\psi)$ of $N$ is an integral submanifold of a distribution $\mathcal{D}$ on $N$ if $$d\psi(M_m)=\mathcal{D}(\psi(m))$$ for each $m \in M$.

A maximal integral manifold $(M,\psi)$ of a distribution $\mathcal{D}$ on a manifold $N$ is a connected integral manifold of $\mathcal{D}$ whose image in $N$ is not a proper subset of any other connected integral manifold of $\mathcal{D}$
Hint: Solve the differentiable equations

$(d/dt x,d/dt y,d/dt z) =(0,-z,y)$. You have:

$x$ is a constant, ${{d^2}\over{dt}}y =-y$ ${{d^2}\over{dt}}z =-z$, thus the flow is $(x_0,acost+bsint,a'cost+b'sint)$.

The other flow is also of the form $(ucost+vsint,y_0,u'cost+v'sint)$ so the maximal integral amifolds are the torus $T^2$.
You have to be more careful. While calculating the flows is the first step, its helpful to also constrain the flows using a boundary condition (like the starting point). This way, while the solution looks like a torus, if you find the magnitude of the flow you'll find that its a different manifold (decidedly not $T^{2}$)