By 1943 the Army had over 900 AntiAircraft Regiments & assorted battalions. With the demise of the Luftwaffe in the Meditereanian there was an over abundance of these units that were no longer needed for antiaircraft defence so they started to deactivate the regiments so the personel could be used elsewhere. Some of the units were disbanded & many of the men were transfered to replacement depots for reassignment as infantry replacements. Many units were formed into Military Police companys and battalions, others were converted to engineer battalions, especialy after many Engineer Combat units of 5th Army left Italy to join 7th Army for the Invasion of Southern France in 1944 which left 5th Army with a critical shortage of engineers.
In the case of the 337th ECB. they were originated from the 3rd battalion of the 209th Coast Artillery Regiment:
209th Coast Artillery Regiment (Antiaircraft) (Semimobile)
14 MAR 44 Redesignated thus:
3rd Battalion > 335th Antiaircraft Artillery Searchlight Battalion at Bari, Italy
335th Antiaircraft Artillery Searchlight Battalion
01 MAY 44 Activated at Bari, Italy
07 DEC 44 Redesignated as 337th Engineer Combat Battalion at Malpensa, Italy.
The 337th was assigned to 1108th Engineer Combat Group, given 30 days training on baily bridges, & mines, then sent out to where the action was.
The 1108th was under IV Corps command.