A method for flying an aircraft along a trajectory subject to time-of-arrival constraints during a cruise phase, the method comprising: if a required time-of-arrival speed mode and cruise optimization are both active (64, 78) and either a vertical profile of an optimum trajectory would significantly improve the fuel efficiency or a required time-of-arrival at a waypoint is unachievable: determining an optimum trajectory that includes a speed schedule that meets the required time-of-arrival at the waypoint and a vertical profile that optimizes fuel efficiency (86, 88-92), the vertical profile being optimized based on a nominal cost index set by a flight crew; and flying the aircraft along the optimum trajectory during the cruise phase, wherein step comprises: (a) determining a first optimum vertical profile for a maximum cost index; (b) calculating a first estimated time-of-arrival for the first optimum vertical profile; (c) determining a second optimum vertical profile for a minimum cost index; (d) calculating a second estimated time-of-arrival for the second optimum vertical profile; (e) determining whether the required time-of-arrival is outside of an estimated time-of-arrival window bounded by the first and second estimated times-of-arrival or not; (f) if the required time-of-arrival is outside of the estimated time-of-arrival window, determining a degraded optimum trajectory that includes a speed schedule that meets the required time-of-arrival at the waypoint and a degraded optimum vertical profile that improves fuel efficiency; and wherein step comprises: (g) if the required time-of-arrival is outside of the estimated time-of-arrival window, flying the aircraft along the degraded optimum trajectory during the cruise phase, wherein step is an integrated function executed by a computer, the integrated function consisting of a required time-of-arrival functionality and a cruise optimization. if a required time-of-arrival speed mode and cruise optimization are both active (64, 78) and either a vertical profile of an optimum trajectory would significantly improve the fuel efficiency or a required time-of-arrival at a waypoint is unachievable: determining an optimum trajectory that includes a speed schedule that meets the required time-of-arrival at the waypoint and a vertical profile that optimizes fuel efficiency (86, 88-92), the vertical profile being optimized based on a nominal cost index set by a flight crew; and flying the aircraft along the optimum trajectory during the cruise phase, wherein step comprises: (a) determining a first optimum vertical profile for a maximum cost index; (b) calculating a first estimated time-of-arrival for the first optimum vertical profile; (c) determining a second optimum vertical profile for a minimum cost index; (d) calculating a second estimated time-of-arrival for the second optimum vertical profile; (e) determining whether the required time-of-arrival is outside of an estimated time-of-arrival window bounded by the first and second estimated times-of-arrival or not; (f) if the required time-of-arrival is outside of the estimated time-of-arrival window, determining a degraded optimum trajectory that includes a speed schedule that meets the required time-of-arrival at the waypoint and a degraded optimum vertical profile that improves fuel efficiency; and wherein step comprises: (g) if the required time-of-arrival is outside of the estimated time-of-arrival window, flying the aircraft along the degraded optimum trajectory during the cruise phase, wherein step is an integrated function executed by a computer, the integrated function consisting of a required time-of-arrival functionality and a cruise optimization. (a) determining a first optimum vertical profile for a maximum cost index; (b) calculating a first estimated time-of-arrival for the first optimum vertical profile; (c) determining a second optimum vertical profile for a minimum cost index; (d) calculating a second estimated time-of-arrival for the second optimum vertical profile; (e) determining whether the required time-of-arrival is outside of an estimated time-of-arrival window bounded by the first and second estimated times-of-arrival or not; (f) if the required time-of-arrival is outside of the estimated time-of-arrival window, determining a degraded optimum trajectory that includes a speed schedule that meets the required time-of-arrival at the waypoint and a degraded optimum vertical profile that improves fuel efficiency; and (g) if the required time-of-arrival is outside of the estimated time-of-arrival window, flying the aircraft along the degraded optimum trajectory during the cruise phase, wherein step is an integrated function executed by a computer, the integrated function consisting of a required time-of-arrival functionality and a cruise optimization.