Maxwell-Boltzmann distribution - find error in derivation I have a derivation of the Maxwell-Boltzmann distribution:

Consider a gas consisting of only one type of molecules, which is in
  an equilibrium with a heat reservoire of temperature T.
Since all three room directions $x,y,z$ are equivalent, the probabilty
  distribution has to be the same for every room direction. The system
  is also invariant under arbitrary reflections. Thus the probability
  distribution only depends on the absolute value of the velocity. Hence
  $p_i(v_i)=f(v_i^2)$, $i=x,y,z$ and a yet to be determined function
  $f$.
Since the velocities in each room direction are independet, we have
  $p(v_x,v_y,v_z)=f(v_x^2)f(v_y^2)f(v_z^2)$ and since the system is
  invariant under rotations, the velocity distributions have to be
  independent, too. Now we can rotate the velocity vector onto the
  $z$-axis and obtain
  $f(v_x^2)f(v_y^2)f(v_z^2)=f(0)f(0)f(v_x^2+v_y^2+v_z^2)$. But this
  property only holds for an exponential function, $f(x)=Ae^{-\alpha x}$, which yields the parametric form of the Maxwell-distribution
  $p(v_i)=Ae^{-\alpha v_i^2}$.

This derivation does not make direct use of the Hamiltonian of the system and I am supposed to find out at what part it is faulty. I.e. the derivation uses an assumption that does not hold in general. 
Now I think that we cannot assume that the physical system is invariant under arbitrary reflections as for example the spins of fermions are not invariant under reflection, but I am not sure with this.
Can anyone help me with this? Thanks!
 A: 
This derivation does not make direct use of the Hamiltonian of the system and I am supposed to find out at what part it is faulty. I.e. the derivation uses an assumption that does not hold in general.

This part seems suspicious:

Since the velocities in each room direction are independent, we have
   $p(v_x,v_y,v_z)=f(v_x^2)f(v_y^2)f(v_z^2)$

In other words, it is assumed that the distribution function of three variables $v_x,v_y,v_z$ factorizes into product of three distribution functions, each being a function of one argument only.
This seems to be a special condition that may not hold for every kind of system. Apparently, it holds for systems that have Maxwellian velocity distribution. I think it may not hold for system of gravitationally interacting particles.
It seems to me that the proper direction in which the above relation of factorizability to Maxwellian distribution should be used is to show that gases obeying Maxwellian distribution (derived based on the Hamiltonian and the Boltzmann distribution) have independent velocity components. 
See also Physical intuition for independence of components of velocity in derivation of Maxwell–Boltzmann distribution
