Why we have public Cat() {} empty?

that is the constructor. If you declare variables in the constructor, all instances which are cat1 and cat2 will have the same value. I think so:))

It is not a function nor a method.

public Cat() {}

notice that there is no return type so this is a constructor. When you provide no constructor for your class, java will create a default constructor. (https://docs.oracle.com/javase/specs/jls/se7/html/jls-8.html#jls-8.8.9) So if you don't create this constructor actually java will insert this:

public Cat( ) { super(); }

my guess is that the creators of this task don't want that behaviour somehow :)

In this case method "Cat() " does nothing. And, it also works - to do nothing :) What number this task?