A valid question! Engine on Linux is dependent on Mainwin libraries. Mainwin does not work on Ubuntu, unfortunately. In June, there will be a new product released called The ArcGIS Runtime, which does not depend on Mainwin, and will be supported on Ubuntu LTS .
ArcGIS Engine for Linux is still not supported on Ubuntu, nor do I know of plans to do so in the future. ArcGIS Runtime for Linux is supported on Ubuntu 10.04 and 12.04.