If you're using an enum, the JVM guarantees only one instance of each value gets created. So in this example, only one instance of MyEnum is ever created however you try and refer to it.
Try printing out m1 == m2 etc if you don't believe that.
By the way, as things stand your MyEnum won't actually compile. You don't need a getInstance() method (just refer to MyEnum.INSTANCE as you have later). But if you want it for some reason it should be static, it shouldn't return int, and it should just return INSTANCE rather than this.INSTANCE. (Oh yes, and INSTANCE should be spelled correctly ).
Out of interest, are you really migrating to Java 1.5? Java 6 came out nearly 6 years ago, and Java 7 a year ago. If you're going to the trouble of upgrading, you might as well go to a current platform.