On May 15, 2016, about 1430 central daylight time, a Piper PA-32R-301 airplane, N185DX, was substantially damaged when it impacted terrain following a precautionary landing 1 mile east of the Alpine-Casparis Municipal Airport (E38), Alpine, Texas. The private pilot and passenger were not injured. The personal flight was conducted under the provisions of 14 Code of Federal Regulations Part 91. Visual meteorological conditions prevailed and no Federal Aviation Administration (FAA) flight plan had been filed for the flight. The flight was originating at the time of the accident.According to the pilot, he had difficulty starting the engine after refueling. He followed the flooded engine start procedures and "ran the engine up" and checked the magnetos. The rpms "seemed satisfactory." The takeoff roll from runway 23 (5,018 feet by 60 feet, asphalt) required "a good bit of runway." During the climb the pilot noted that the exhaust gas temperature (EGT) was "pegged" and he reversed course to return to the airport. He stated that he was not able to find a "responsive power setting" and adjusted the mixture in attempt to cool the engine. He stated that the engine was not "missing." He left the landing gear up and after the airplane cleared powerlines the pilot "cut the engine." He performed a gear-up precautionary landing at which time the right wing and fuselage sustained substantial damage. The engine was examined under the auspices of the National Transportation Safety Board investigator-in-charge. The number six spark plug exhibited signatures consistent with a lean mixture. The remaining lower spark plugs and the engine cylinders contained carbon deposits consistent with a rich fuel/air mixture. The flame tubes within the muffler were cracked and deteriorating but remained in place. The EGT was connected to the engine via a single probe attached to the number six cylinder. The connection was secure and the probe cable was unremarkable. An external examination of the engine revealed no mechanical anomalies that would have precluded normal operation. The engine remained on the airframe. The airframe was secured to a trailer and an external fuel supply was connected to the engine. The propeller was impact damage and exhibited rotational scoring and bending on each blade. For the examination and test run it was replaced with a test propeller. The engine started without hesitation and ran for 15 to 20 minutes at varying power settings. The magneto check and propeller governor check were normal and 2,700 rpm and 29 to 30 inches of manifold pressure were obtained. The investigator noted that the EGT was immediately high (near or above the peak EGT) and fluctuated at varying power and mixture settings. At the higher power settings the EGT increased to the maximum reading and then fell off to zero. The engine was shut down and examined. All six spark plugs were darker in color consistent with a rich fuel/air mixture. The number six fuel injector was found partially occluded and the obstruction was removed. A second engine test run was conducted about two hours later. The EGT continued to fluctuate at lower power settings and was generally high (near or above the peak EGT). At higher power settings the EGT increased to the maximum reading and then fell off to zero. An examination of the engine and related systems revealed no mechanical anomalies that would have precluded normal operation. Calculations of relevant meteorological data revealed that the density altitude at the time of the accident was 5,750 feet.