How can I have a Singleton that's derived from an abstract base type in Java?

I have some classes that are used as Singletons. They share some basic functionality and extend the same ancestor from a library which in turn isn't normally used as a Singleton.

If I put the common functionality in a base class that inherits from the common ancestor, I get a class which makes no sense to instantiate, so I made it abstract. Also, because the classes are all used as Singletons, they should all have an init() and a getInstance() method, which are both static. All the constructors are of course non-public.

Now, since static is an illegal modifier for abstract methods, the following doesn't work, although this would be exactly what I want:

class Base extends LibraryClass {
    protected Base() {
        // ... constructor
    }

    // ... common methods

    // ILLEGAL!
    public static abstract void init();
    public static abstract <T extends Base>T getInstance();
}

class A extends Base {
    private static A _INSTANCE;

    private A() {
        super();
    }

    public static void init() {
        _INSTANCE = new A();
    }

    public static A getInstance() {
        return _INSTANCE;
    }
}

I could just leave out the illegal lines in the base class and be done with it. But how do I express that every child of Base must have these methods?