Summary: The main result ensures that the scalar problem \(x''= f(x)\), \(x(0)= x_0\), \(x'(0)=x_1\), has a nonconstant locally \(W^{2,1}\) solution if and only if there exists a nontrivial interval \(J\) such that \(x_0\in J\), \(f\in L^1_{\text{loc}}(J)\), \(x^2_1+ 2\int^y_{x_0} f(s)\,ds> 0\) for almost all \(y\in J\) and \[ {\max\{1,|f|\}\over \sqrt{x^2_1+ 2\int^._{x_0} f(s)\,ds}}\in L^1_{\text{loc}}(J). \] Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.