What we commonly call snow-on-the-mountain is really bishop's weed or goutweed (Aegopodium podagraria 'Variegatum'). It is a nice ground cover for shade and grows very vigorously. The green and white foliage is what is called a "sport." A sport is a mutation that takes place in plants from time to time and is sometimes propagated because it's interesting. Occasionally, the plants revert back to a solid form. You can remove the green plants and maintain only the variegated ones. Because they can be transplanted easily, you could fill the area that the green ones occupied with transplanted variegated ones.