Singular locus on the space of genus two curves with split Jacobians

We study the singular locus on the space of genus 2 curves with a (3, 3)-split Jacobian. Such space was computed by Shaska in [3] where it was the first time that an explicit equation had been given for the space of genus 2 curves with (n, n)-split Jacobian, for n odd. We sompute the singular locus of such space and study the arithmetic proporties of some of the curves in this locus which are defined over the rationals.