Changes to video encoding program in C that uses FFMPEG (change the encoder codec used)

I have a working C program that uses FFMPEG and x264 to transcode any video file to a h264 output file. The program takes a video file as an input, creates a new video output container, then reads the existing file frame by frame and encodes these frames one by one to the output file, using x264 encoder. For licensing reasons I am looking to move away from x264 so that I am able to distribute the software freely. I therefore need someone to suggest an alternative output encoder that would be free to use and distribute inside my software, and then make the change to my program to use this encoder instead of x264. The project would include three steps: - Familiarise yourself with the existing code and how it works (quite simple) - Modify it to use a different encoder - Test it with different input formats making sure the output file works fine