Singleton Pattern with a public constructor

I want to implement Singleton Design Pattern to a class which extends another class with a public constructor in PHP. For example, consider a class, say Animal:

class Animal
{

    public function __construct()
    {

    }
}

I want to implement Singleton Design Pattern to a class which extends Animal, for example:

class Dog extends Animal
{

    protected function __construct()
    {
    }

    public static function getInstance()
    {
        static $instance = null;
        if (null === $instance) {
            $instance = new static();
        }

        return $instance;
    }
}

If I try to do Dog::getInstance() it gives an error

Fatal error: Access level to Dog::__construct() must be public (as in class Animal)

But if I make the constructor of class Dog public, then there's no point in implementing Singleton pattern to it.

Is there a workaround so that I can implement Singleton pattern to the class Dog while extending class Animal? OR Is there a better design pattern for achieving what I want?

Thanks.