  In this work, we study the $k$-median clustering problem with an additional
equal-size constraint on the clusters, from the perspective of parameterized
preprocessing. Our main result is the first lossy ($2$-approximate) polynomial
kernel for this problem, parameterized by the cost of clustering. We complement
this result by establishing lower bounds for the problem that eliminate the
existences of an (exact) kernel of polynomial size and a PTAS.
