A combustion system, and method. A combustor oxy-fired at high pressure delivers flue gas at pressure to a condensing means, such as a condensing heat exchanger, to produce a high temperature condensate for delivering thermal energy to an industrial process system, particularly for power generation, including a Brayton cycle, a Rankine cycle, or a binary fluid cycle system such as a Kalina cycle, and in particular as a bottoming cycle for an organic Rankine cycle. The combustor can concurrently provide direct heat to a secondary system, including a Brayton cycle system, a Rankine cycle system, and a binary fluid cycle system such as a Kalina cycle, without requiring significant modifications to the secondary system. The system and method provide for efficient and advantageous use of the higher temperature condensate produced.