I am working on stability of nonlinear switched systems and recently, I have proven that switched systems with homogeneous, cooperative, Irreducible and commuting vector fields , i.e., vector fields with Lie bracket equal to 0, are Dstable under some condition. I was trying to find an example for such systems but surprisingly, I could not find any in the papers which have dealt with them. Does anybody know a good example of commuting nonlinear vector fields? And are there any significance to such systems (from physical point of view)?

Any set of nonlinear coordinates gives you a corresponding set of commuting vector fields. So any of the coordinates listed in the "See also" section of http://en.wikipedia.org/wiki/Elliptic_coordinates gives an example. 


You can take any two commuting vector fields, e.g. $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ in the plane, then conjugate by a diffeomorphism. The resulting two vector fields will commute, but need not be linear (at least if I understand what you call linearity of a vector field). 

