The direction of electric field is perpendicular to the equipotential surfaces. Here the electric field is in $+z$ direction. And the electric field is always remain in the direction in which the potential decreases. Its magnitude is given by the change in the magnitude of potential per unit displacement normal to the equipotential surface at the point.
So, electric field in z-direction suggests that equipotential surfaces will be the plane perpendicular to z-axis along $x-y$ plane. Therefore the potential is a constant for any $x$ for $a$ given $z$ and for any $y$ for a given $z$ and on the $x-y$ plane for a given $z$.