You sure can use your own personal license on another machine, as long as it is your license.
The SlickEdit End User Agreement specifically allows for that regarding your SlickEdit for Linux Named User license:
(i) Named User License. Software provided to You under this option may only be installed on any computer owned or controlled by You, but in no event shall the Software be (i) used by any person other than You and (ii) used on more than one computer at the same time.
Hope this information helps.
SlickEdit Support