Maximization of sum of a bivariate optimization problem

Question

Assume $f(x+y):\mathbb{R}\rightarrow \mathbb{R}$ is strictly increasing and concave in the sum $z=x+y$. Observe the bivariate maximization problem $$ \underset{x,y}{\operatorname{max}} J(x,y)=f(x+y)-\frac{x^2}{2}-\frac{y^2}{2} \tag{1}. $$ The first-order conditions $$ \frac{\partial J(x,y)}{\partial x}=f'(x+y)-x=0,\tag{2} $$ $$ \frac{\partial J(x,y)}{\partial y}=f'(x+y)-y=0,\tag{3} $$ Jointly determine the solution to this bivariate optimization problem. Moreover, from $(2)$ and $(3)$, we know that for a maximum, it must be the case that $$x=y.\tag{4}$$ Now I am primarily interested in the sum $z$. Adding the two first-order conditions gives an optimality condition for $z$ as $$ f'(z)=z.\tag{5} $$ By symmetry $(4)$ and the optimality condition $(5)$, I now want to define a symmetric objective function $\tilde{J}(y)$, to which $(5)$ is a solution. How do I proceed? Normal integration obviously does not work.