Can a person legally write new song to an existing melody without paying royalty?

It depends on the copyright status of the melody. If the melody is in the public domain, then anyone can write new songs to it without paying royalties. However, if the melody is still under copyright, then you will need to obtain permission from the copyright holder before you can legally use it.

There are a few ways to determine if a melody is in the public domain. One way is to check the copyright date. If the melody was published before 1923, then it is likely in the public domain. Another way to check is to see if the melody is listed in the U.S. Copyright Office's catalog of public domain works.

If you are not sure whether a melody is in the public domain, it is best to err on the side of caution and obtain permission from the copyright holder before you use it.