The authors study for \(a>0\) and \(b>0\) the functional equation \[ af(x)+bf(y)=f(ax+by)g(y-x),\qquad x,y\in \mathbb R, \] where the functions \(f,g:\mathbb R\to \mathbb R\) are assumed to be locally integrable and continuous at the origin, respectively. They find the solutions under various assumptions on \(f(0)\) and \(g(0)\) and the derivatives of \(f\) and \(g\) at the origin. A major distinction is between the cases \(a+b=1\) and \(a+b\neq 1\). For the special case \(a=b=1/2\) they refer to \textit{Z. Powązka} [Rocznik Nauk.-Dydakt. Pr. Mat. 15, 119--128 (1998; Zbl 1160.39314)].