On the functional equation x + f ( y + f ( x )): y + f ( x + f ( y )), II

For an abelian group (G,+,0) we consider the functional equation 1 $$f : G \to G, \,\, x + f(y + f(x)) = y + f(x + f(y)) \quad (\forall x, y \in G),\qquad \qquad\quad (1)$$ f : G → G , x + f ( y + f ( x ) ) = y + f ( x + f ( y ) ) ( ∀ x , y ∈ G ) , together with the condition 0 $$f(0) = 0. \qquad\qquad\qquad\qquad\qquad (0)$$ f ( 0 ) = 0 . The main question is that of existence of solutions of (1) $${\wedge}$$ ∧ (0), specifically in the case when G is the direct sum $${\mathbb{Z}_n^{(J)}}$$ Z n ( J ) of copies of a finite or infinite cyclic group (Theorems 3.2 and 4.20)