Is it possible to implement the "Singleton" design pattern using a non-private constructor?

Question

C++ / C#

Interested in the possibility of implementing this pattern in C # without a private constructor
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Expanded the question.
Is it possible to create a condition inside the constructor that would prohibit the creation of objects
Like this:

class Singleton
{
    private int countInstance;
    public Singleton()
    {
        lock (syncRoot) {
        if (!countInstance == 0) 
        { // код, запрещающий создание экземпляра 
        }
        else 
        {
            countInstanse = 1;
        }
        }
    }
}

Answer 1

No, the constructor is made private so that only the class itself has access to the constructor. Otherwise, anyone can create a singleton.