Summary: The author establishes existence conditions for solutions bounded on the whole real axis to weakly nonlinear ordinary differential equations under the assumption that the linearized problem is characterized by a Noether operator \(L\) and has an \(r\)-parametric family of solutions bounded on \(\mathbb R\). In case when the operator \(L\) is Fredholm and \(r=1\), the well-known Palmer’s result is obtained.