A cone has all the same local geometrical properties as a plane, so if you take a piece of graph paper and form it into a cone, [itex]\partial_x[/itex] and [itex]\partial_y[/itex] still satisfy the Killing equation. On the other hand, the cone has intrinsic geometrical properties that are different from those of the plane, e.g., parallel transport around a loop enclosing the tip will cause a vector to rotate. This singles out the tip and gives it a special geometrical role, which is clearly not consistent with translational symmetry. Does this mean that we can have a field that satisfies the Killing equation without being a Killing vector, or is the Killing equation violated at the tip of the cone? Does it matter if you extend the cone to make a double cone, so that orbits of a Killing vector can pass smoothly through the tip?