That isn't a static connection - with a static connection there is only one per application, and you don't want that.
What you should be doing is creating your connection, opening it, using it, and disposing of it when you are finished - the easiest way to do that is with a using block:
using (SqlConnection con = new SqlConnection(strConnect)
{
con.Open();
... use it here ...
}
The system will then Close and Dispose the connection object for you automatically when it's no longer available. You should also be doing this for SqlCommands, SqlDataReaders, SqlDataAdapters, and so forth.
This prevents some of the nastier problems that crop up later, such as a DataReader being accidentally being left "open" on a connection - which prevents anything else being done with it later.
If you need a connectiuon to another server / DB, you just create that Connection when you needit inside another using block and let the system handle it.