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2)Have you come across this error for Authenticated users?
Failed to create a list item for this external list based on the Entity (External Content Type) 'ECT' in EntityNamespace '/sites/test space'. Details: Credentials were not found for the current user within the target application 'ECT'. Please set the credentials for the current user.
3)Edit the external Content Type,and make sure in Connection properties ,Authentication mode is selected to 'BDC Identity'