EMB-505 pilot reported an engine failure at cruise altitude; and inflight engine shutdown. The flight crew successfully diverted and landed.

Narrative

About 10 minutes after takeoff from ZZZZ; around FL280; the aircraft shuddered violently; a loud bang was heard and a burning smell was perceived. We then received a series of CAS messages followed by a failure of Engine 1. We leveled off at FL300; and notified ATC that we had an engine failure and [advised ATC]. We ran the QRC and QRH for Engine Failure in Flight and secured Engine 1. We decided to divert to ZZZ; which was directly ahead; but then were advised that the weather was low IFR; so we rediverted to ZZZ1. After contacting Company; we considered running the Inflight Engine Restart QRH; but N2 was indicating 0% RPM; so we did not attempt a restart. We then began to descend and prepared to land at ZZZ1. We selected the RNAV (GPS) approach for Runway XXR; and headed for the IAF. As we descended; we noted that N2 had climbed to 4%. ATC asked if we needed assistance on landing and we declined because we had no indications of fire or gear failures. ARFF still came out to watch us land. We coordinated with ATC and FBO ZZZ1 to tow us to Customs after we cleared the runway at [Taxiway] XX. On postflight I could not see any damage; but the oil level was not visible in the sight gauge.