Set up a US branch of my Luxembourg company?

I run a small Luxembourg-based consulting company and I'm about to start organizing events such as short seminars (2 hours) to long workshops (up to 5 days).

I'd like to tour the USA and possibly Canada with my seminars and workshops also, delivering, say, 60 to 100 days of seminars/workshops per year in the US.

Would it be advisable to set up a US branch of my company? I would of course be on the road whenever I'm in the US, and most of the time I'd in Europe. I'm also not a resident alien. So I suppose another question would be: as a non-US citizen, non-resident alien, can I set up shop in the US?

The issue might be more of an immigration than of business organization. An alien can definitely own a business in the US. The problem is that an alien cannot work in it. Coming for a single 2 days workshop may be fine under B1 visa. I'm not sure if coming under B1 will be fine for a recurring 100-days-a-year employment. You need to consult an immigration law attorney in the US on that (and same for Canada).

As to the business organization per se - you don't have to set up a branch in the US, but you might be required to register your company as a foreign entity in the specific States you'll be operating in (in the US - the States register companies, not the US government). Your company might also be required to pay taxes on the US-derived income (as you personally, too) to the IRS, and to each State for the income derived in that State.