Suppose one is given a set of vector fields $X_{\alpha} \in \mathbb{R}^n$. Is it generally possible to construct a manifold embedded in $\mathbb{R}^n$ going through a particular point and whose tangent space is spanned by the $X_\alpha$? Will that manifold be unique?
I can imagine constructing this using something similar to Forward Euler which converges (for sufficiently nice functions). And perhaps what I am asking for is in fact just the flow of the vector fields. If it is in general possible to construct such a manifold, is there a general constructive method for doing so?
To be specific this comes from the following question: Given the vector fields (in $\mathbb{R}^3$) $$ X_1 = x \partial_y - y \partial_x,\quad X_2 = z\partial_x - x\partial_z,\quad X_3 = z\partial_y - y\partial_z, $$ what is the manifold in $\mathbb{R}^3$ that includes the point $(x,y,z) = (1,0,0)$ and whose tangent space at each point is spanned by the above vector fields? I can easily show that $S^2$ does in fact satisfy all of the above conditions. However, (a) I do not know that it is unique (b) I use quite a lot of guessing and checking to get there. I imagine there is a system of PDEs that can be written down for the coordinate charts whose solution gives coordinates for the sphere, but all of my attempts have not resulted in anything that looks like coordinates for a sphere. By examining the span of the vector fields at $(1,0,0)$ I can see that the manifold must be a graph in $\mathbb{R}^3$ so I write it $g(x,y,z)=0$. Then the above vector fields give that $g$ must satisfy $$ x \partial_y g = y\partial_x g, \quad z \partial_y g = y\partial_z g, \quad z\partial_x g = x\partial_z g $$ which (as I can check) is satisfied by $g(x,y,z) = x^2 + y^2 + z^2 -1$ but I cannot show gives rise to such a $g$.