Working in Germany, where Gauss_Kruger is very often used (as well as ETRS1989_zone 32N), and trying to find out about WKID when I realised that on my machine, the projection reference of the DHDN_3_Degree_Gauss_Zone_3 is set on Transverse Mercator, whereas for Gauss_Kruger 1,2,4 or 5, it's set on Gauss_Kruger, which makes sense.
I have compared with the Parameters on other machines, where they are correct (Gauss_Kruger). Somehow, I have changed them and I need to modify, and not by setting a new custom Coordinate System but by modifying the file within ArcGIS or even, deleting it. Now I read things there is no Coordinate System Folder but " Instead, a virtual directory structure is built directly from the coordinate system data." (http://support.esri.com/en/knowledgebase/techarticles/detail/39488)
How can I do this?