Can you lose 10 kilos in 2 to 3 weeks in a healthy way?

No.

Unless you're having a sizable tumour removed, then that's too fast. The loss will be mostly muscle, with some water and waste products. You will go back up once your normal diet resumes. Depends. If you are a person who weighs between, say, 50 k and 100 k, the answer is: no, absolutely not. However, if you are significantly overweight, bedridden, for example, then yes, it could be possible to do so in a "healthy way". Would losing that much weight that quickly be healthy - probably not, even if you are obscenely overweight - being that heavy creates its own unique health issues, losing that much weight in that short a period of time might send a person into shock.