cosmic conspiracy for rotation curves

What’s the density distribution inside a dark matter halo that allows flat rotation curves? How does that correspond to the NFW-profile shape and what’s the reason that the size of the galactic disk is roughly equal to the NFW scale radius?

bonus question: Can you show using a power series that \(\exp(A)\exp(B)=\exp(A+B)\) for two square, real-valued, commuting matrices \(A\) and \(B\), i.e. \(\left[A,B\right]=0\). What changes if the two matrices are non-commuting?