Boundary value problems of the form $x'' = -\lambda f(x^+) + \mu g(x^-)$ $(i),$ $x'(a)=0=x'(b)$ $(ii)$ are considered, where $\lambda,\mu>0$. In our considerations functions $f$ and $g$ may be nonlinear. We give description of a solution set $F$ of the problem $(i)$, $(ii)$~- a set of all triples $(\lambda,\mu, \alpha)$ such that $(\lambda,\mu,x(t))$ nontrivially solves the problem $(i)$, $(ii)$ and $|x'(z)|=\alpha $ at any zero of $x(t)$ $(iii)$. It turns out
that the solution set $F$ is a union of solution surfaces
$F_{i}^{\pm}$ $(i=1,2,\ldots)$ and the non coinciding solution
surfaces are centro-affine equivalent. Cross sections of the
solution surface $F_{1}^{\pm}$ with planes $\alpha=const$,
$\mu=const$ and $\mu=const\cdot \lambda$ will be analyzed for the
nonlinearities $f=g=x^3+x^{41}$.