Contact a plumber to clear any internal drain lines and to clear the private sewer line from your home to the connection on the Authority's main line. Once the private sewer line is cleared to the Authority's main sewer line and all roots, grease, and debris have been removed, then the private line should drain as normal. This should be done before you contact the Authority to investigate the issue.
Homeowners own the private sewer line from the house to the Authority's main sewer line. As such, they are responsible for maintenance and repair of this private sewer line. All costs involved with locating the problem are the responsibility of the property owner.