Is it legal to integrate the Ubuntu system in my software product?

Software within Ubuntu is widely GPL compatible and various essential base components are certainly GPL2 or GPL3. Those are the licenses you're going to have to abide by.

The core understanding is that if you include this software, you need to make the source code available. I'm not sure if that stretches to your software (if it just sits on top) but if you've customised any of the Ubuntu system by modifying existing files, you must provide source for that.

I would strongly suggest talking to a professional about this. The FSF might be able to help but either way, I would start just by reading through what the GPL is and how it effects you:

• http://www.gnu.org/licenses/
• http://www.gnu.org/licenses/gpl-faq.html

People take these licenses seriously. Don't ignore them.

Just to express this again: GPL is not the only license in use (there are a myriad of various open source licenses in use in the default install of Ubuntu) but adhering to GPL should (talk to a lawyer about how you do that and to indemnify yourself that if this is true) mean you're also complying with their licenses. Aka, GPL is the fussiest license.

If you install other things (including things from the repo, or software that Ubuntu offers to install like drivers), these may have different licenses and you'll need to audit that. See the comments for some questions on dealing with that aspect. The rest of my answer assumes a clean Ubuntu install.

Refer to the Ubuntu licensing information. If you are still unsure it would be best to email Canonical to confirm. It seems to depend on individual packages involved and their licences, so you may need to read each piece of software's license you use to check that they do not rule out commercial uses.