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Most of the Minnesota county coordinate system prj files in ArcGIS have incorrect values. For many of the zones based on the Lambert Conformal Conic projection, the latitude of origin parameter values are incorrect. The zones based on Transverse Mercator have incorrect false easting and false northing parameter values.
The prj files for these counties are correct:
Lake of the Woods North
St. Louis (North, Central, South)
The prj files for the Minnesota county geographic coordinate systems are correct.
The projected coordinate system definitions were defined incorrectly.
Replace the existing files with corrected versions.
The Coordinate System directory is located in the ArcGIS installation directory.