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I have created an application pool called "schoolPool" and assigned it to my web application. Identity for this pool has been set to LocalSystem. When I try to access my database from within the application, i.e. open a SQL connection, I get the following error all the time:
Login failed for user 'NT AUTHORITY\SYSTEM'
I tried to add NT AUTHORITY/SYSTEM to SSMS (SQL Server Management Studio) logins, but it was already a principal, showing the following error: 
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To summarize, The Login failed for user 'NT AUTHORITY\ANONYMOUS LOGON' error in both our cases seems to be caused by a service not running and/or not on the right user. Ensuring the right SPN or other Service is running and under the correct user should solve the anonymous part of the problem.
The account NT AUTHORITY\System which is a Local System account.. It is a powerful account that has unrestricted access to all local system resources. It is a member of the Windows Administrators group on the local computer, and is therefore a member of the SQL Server sysadmin fixed server role.
The generic message “Login Failed for User (Microsoft SQL Server, Error: 18456)” means you entered invalid credentials when logging into SQL Server.
Allow NT AUTHORITY/SYSTEM to server Role as sysadmin. 
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