neither, nor and not ... either

1) neither and nor

We can use neither and nor as adverbs to mean 'also not'. Neither and nor come at the beginning of a clause, and are followed by inverted word order

auxiliary verb + subject.

I can't swim. < Neither/nor can I. (NOT [also can't.)
Ruth didn't turn up, and neither/nor did Kate. (NOT ... and Kate didn't too.)

In American English, nor is not normally used after and.

2) not either

We can also use not ... either with the same meaning and normal word order.

I can't swim. ~I can't either.
Ruth didn't turn up, and Kate didn't either.

In very informal speech, me neither (and occasionally me either, especially in AmE) can be used instead of I ... n't either.

I can't swim. ~Me neither.

3) one negative

Only one negative word (not or neither) is necessary to give a negative meaning.

Neither did Kate OR Kate didn't either. (NOT Neither didn't Kate OR Kate didn't neither).

Michael Swan, Practical English Usage, Third Edition, Oxford University Press, 2006.