Let us evaluate the original sentence. On the first reading, the sentence does not sound very convincing.
(A) all layers of the earth are contained solely
This is incorrect because "all of the water" is singular. Hence are is incorrect.
(B) all layers of the earth are contained.
Incorrect because of the same reason as A.
(C) all layers of the earth is contained solely
This sentence makes a redundant usage, "solely" becomes redundant because of the phrase "all of the water".
(D) all layers of the earth is contained
This sounds correct, let us hold on to it. It eliminates the error of singular/plural and solely.
(E) many layers of the earth is contained
The usage of many layers changes the meaninig of the sentence.
During the exam, the recommended strategy would be to use 3-2 splits.
We can see the split between usage of "are" & "is". As discussed above, the correct usage will be "is". This leaves us with C,D,E.
Now the split is 2:1 the usage of solely. The usage of solely is incorrect as discussed above because of redundancy. Eliminate C.
We are left with D,E. The difference between the two is all vs many. The usage of many changes the meaning of the sentence. We are left D.