I have a PR in Canada and have been living/working in Toronto for the past 15 months. I have another 3 years in which I have to accumulate 9 months of stay in Canada in order to meet the residency requirements in Canada. Meanwhile, I have a full scholarship offer from a US university to pursue PhD. I got my US F1 (student visa) approved. In order to maintain my residency in Canada, I plan to travel to Toronto every month for 4-5 days while I'm studying in the US. I have an offer to do some private consulting from a business that has offices both in Canada and US.

Since the F1 visa in the US, prohibits doing any work when physically present in the US, I plan to work for this business while I'm visiting here those 5 days in the month. I'll be paid in Canada. I believe, from an immigration perspective it is acceptable as long as I do not work while physically present in the US. is that right?

What are the US/Canada employment legalities in this situation for me AND my employer. Any advise will be greatly appreciated.