In the two body problem we have two particles interacting via a potential $V(| r_1 - r_2 |)$: $$ H = \frac{p_1^2}{2m_1} + \frac{p_2^2}{2m_2} + V(|r_1 - r_2|) \, .$$ It is well known that, with a canonical transformation, the latter can be rewritten as $$ H = \frac{P^2}{2M} + \frac{p^2}{2\mu} + V(|r|) \, ,$$ where $P = p_1 + p_2$, $M = m_1 + m_2$, $ \frac{1}{\mu} = \frac{1}{m_1} + \frac{1}{m_2}$. This is the Hamiltonian of a free particle and a particle subject to a central potential.
Question: Is it possible to apply this scheme if the initial particles (1 and 2) are identical (bosons or fermions)? How can we symmetrize / anti-symmetrize the resulting wave function?