US/Canada employment question

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. From an immigration perspective it is acceptable as long as I do not work while physically present in the US.

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

It is okay if you are out of the country and paid out of the country and taxed as to withholdings there as well.

...but I would still get an immigration lawyer to approve this plan after reviewing your details. Just to be safe.