The "we are closing our doors" notice on the DynamixMusic site says that they have "a large selection of albums and singles available on Apple Music" – and as of right now, it looks like there are Dynamix tracks available in the iTunes Store (for purchase) and on the Apple Music streaming subscription service.
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As far as previously-purchased music, are you saying that
- The app downloaded it to its own area (rather than to your Music library), and that
- The music might be in some proprietary and/or DRM-infested format?
If the music is in non-DRMed MP3 or AAC format, but is stored in an area associated with the app (i.e., it isn't in the Music library), I don't know if you might be able to "pry the song files out" with the aid of
- The Files app (on the iPhone)
- The Finder / iTunes / Apple Devices (on a Mac or PC)
- A third-party iPhone management application (like iMazing)
and then import them to a Mac or PC for manual synchronization to the Music library on the iPhone. That might be something you want to ask Dynamix about.
If it is in some proprietary format or has DRM, you might be up the creek without a paddle.