We prove that any maximal group in the free Burnside semigroup defined by the equation xⁿ = x^n+m for any n ³ 1]]> and any m ³ 1]]> is a free Burnside group satisfying x^m = 1. We show that such group is free over a well described set of generators whose cardinality is the cyclomatic number of a graph associated to the J]]>-class containing the group. For n=2 and for every m ³ 2]]> we present examples with 2m-1 generators. Hence, in these cases, we have infinite maximal groups for large enough m. This allows us to prove important properties of Burnside semigroups for the case n=2, which was almost completely unknown until now. Surprisingly, the case n=2 presents simultaneously the complexities of the cases n=1 and n ³ 3]]>: the maximal groups are cyclic of order m for n ³ 3]]> but they can have more generators and be infinite for n £ 2]]>; there are exactly 2^|A| J]]>-classes and they are easily characterized for n=1 but there are infinitely many J]]>-classes and they are difficult to characterize for n ³ 2]]>.