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How center of mass is conserved in the vertical plane ???
One cannot consider the force of gravity to be an internal force while taking the wedge and the body to be a system. So, since an external force of gravity acts on the body , the center of mass can get accelerated.
However if one is taking the earth and the body sliding down the wedge as a sytem, no external force would be acting on the body as gravty would now be an external force . and so the center of mass should not be unaccelerated and so is the case. Actually when the ball slides down the wedge with an acceleration , the earth too gets gets an accereration in the upward direction , however due to large mass of the earth, the acceration is so small that often it is neglected.
How center of mass is conserved in the horizontal plane ???
The body is not experiencing any force in the horizontal direction so the cetre of mass should remain at rest in the horizontal plane. However the body sliding on the wedge is making motion in the horizontal plane too. To account for this fact one can argue that when the body makes motion on the wedge in the horizontal plane the wedge too makes the accereration but in the opposite direction.As such the center of mass remain at rest
Note: However in most case this small acceration of the wedge is cancelled by the friction and the wedge donot get accelerated. However in this case the friction if the external force causing center of mass to accelerate.
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