  Using the navigation data (h,W) of a Kropina space, we characterize
weakly-Berwald Kropina spaces and Berwald Kropina spaces by means of the
Killing vector field W and the parallel vector field W, respectively. Moreover,
the local 1-parameter group of Finslerian local isometries of the Kropina space
coincides with the local 1-parameter group Riemannian of local isometries on
(M, h).
