Three years ago I was working in a public research organisation, and we obtained a patent, I am one of the authors, the research institution payed everything. Later I moved to a different country and institution. Since I am one of the authors of the patent, but I am in another institution, can I exploit commercially the patent?
This is a multi-layered question.
1) An inventor who fully assigns his patent rights to someone else (such as his employer) generally does not have any right to practice the invention. Now, you could have contracted for a license-back, but I suspect that you did not. In addition, you should remember that patents expire early because companies decide not to pay the renewal (maintenance) fees.
2) Patents are territorial. Thus, the fact that a patent exists in the USA has no bearing on whether someone can practice the invention in Canada or Germany. It is expensive to get patents in every country and so there is a good chance that it is not patented in your new country.
3) Former employees should always be concerned with being accused of a breach of confidentiality, trade secret violation, and breach of a covenant not to compete. I do not know your particular situation, but one nice thing is that the patent is public and so there can't be any accusations of trade secret violations for using that invention.
Probably not, but it depends on the law of the country in which you worked. In many places, the invention belongs by law to the employer. It also depends on whether the invention is covered in your current country. I suggest you obtain professional advice before taking any actions.