In　this　paper,　we　investigate　systematically　the　vibration　of　a　typical　2DOF　nonlinear　system　with　repeated　linearized　natural　frequencies.　By　application　of　Descartes＇　rule　of　signs,　we　demonstrate　that　there　are　14　types　of　roots　describing　different　modal　motions　for　varying　nonlinear　parameters.　The　method　of　multiple　scales　is　used　to　obtain　the　amplitude-phase　portraits　by　introducing　the　energy　ratios　and　phase　differences.　The　typical　nonlinear　in-unison　and　elliptic　out-of-unison　modal　motions　are　located　for　the　14　types　of　roots　and　then　validated　by　numerical　simulations.　It　is　found　that　the　normal　in-unison　modal　motions,　elliptic　out-of-unison　modal　motions　are　analogous　to　the　polarization　of　classical　optic　theory.　Further,　some　kinds　of　periodic　and　chaotic　motions　under　out-of-unison　and　in-unison　excitations　are　investigated　numerically.　The　result　of　this　study　offers　a　detailed　discussion　of　nonlinear　modal　motions　and　responses　of　2DOF　systems　with　cubic　nonlinear　terms.