This site explains it all in detail with examples http://hubpages.com/hub/Grammar_Mishaps__Lie_vs_Lay

We use word "lay" when we want to put sb/ smth carefully in a particular position or on a surface. We can use this in that sentences: He is trying to lay the child gently down on her bed. "Don't worry" she said, laying her hand on my shoulder. We can use also lay when we want to put smth in the correct position for a particular purpos. In sentence like this: They're laying new electrocity cables in our street. We use lay when we want to say that hens produce eggs - hens lay eggs or to prepare smth for use - Can you lay the table for a supper, please? ;o) Word LAY is also the word of past tense LIE - means to be in or more into a flat or horizontal position. We use it In sentences like: He lay on the sofa and went to sleep. The book lay open in front of her. We can also use past tense when smth was in a certian state or position - Snow lay thick an the ground. It is incredible in how many situations we can use this word but everything depens on the situation.

The problem has been solved (past event), and the answer was discovered in an observation already made (again, past event). So it makes no sense to use the present tense, "lies in". "Lay in" is past tense. "The key to a solution of that problem used to lie in..." means it doesn't lie there anymore, it now lies somewhere else. This sentence also suggests the problem remains unsolved.