Another is litterally - an other. An, the indefinite article, means "any" as opposed to "the" which is the definite article, and means
one specific.
Another refers to some other choice; and imples more than one.
The other implies that it is the second choice of only 2.
1) We can handle this issue one way or the other.>> There are 2 ways to handle the issue.
2) We can handle this issue one way or another. >> There are 2 ways or more to handle the issue. You don't know all the ways yet.
3) On one hand, he can do this, on the other hand he can study.>> There are only 2 choices - "this" or "study". Imagine you hold a peice of paper in one hand that says "this" and a paper in your other hand that says "study".
4) On one hand , he can do this , on another hand he can study. >> Not correct unless you are an Indian god such as Shiva or Ganesh who have at least 4 arms.