why to study

English as a Second Language English Vocabulary, Grammar and Idioms
#1

“We don’t know why to study English.”
If this sentence is wrong, please help me to re-phrase it.

#2

Yes, this sentence is incorrect. Why, indeed, is not followed by a to-infinitive,
but by a subject +verb.

I think, this sentence can be re-phrased into the following one:

We don't know why [u]we study[/u] English.
#3

Yes, Foreigner I identify with you.

But, the following could also work:

…how to study English.

#4

It would have a completely different meaning.