There is no “generic” way to just smash WP with another app and have it work.
It can be roughly split in two cases:
- Applications run separately on server, installed in different directories, and styled to look as one site (possibly talking to same database, possibly talking to each other via REST API or something).
- One of the applications runs whole site (URLs, routing, templates, styles, etc) and incorporates another in PHP runtime to perform specific functionality.
It is hard to advise on specific case without thorough research into circumstances.
In general WP is a PHP runtime and integrating other PHP solutions with it is perfectly possible.