Does an engine that combines an air-breathing rocket and non-air-breathing rocket exist?
There's a very recent claim for an engine called Fenris, which claims to be a no-moving-parts "rocket" that uses ambient air for its oxidizer.

On the face, that sounds like a ramjet, but the claims made by the inventors appear (to me) to be investor bait, rather than anything that could actually be true.  Video on the entry page at the inventor's web site shows a burn more like that of a poorly designed torch, than a rocket or jet.

As noted in comments, there is also the SABRE engine concept -- this is not yet a working engine, as far as I know, but will/would be a hydrogen-fueled turbo-ramjet which uses the liquid hydrogen to liquefy intake air while in jet mod, for use as oxidizer when it switches to rocket mode.  The intent is for use as a spaceplane-to-orbit.