I would like to execute the Curves And Lines add in from my Python script.
Is there any interface for working with add-in inside of a Python script?
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You do not say where you wish to run your Python script but, in any event, since Python AddIns only work from within ArcMap I think the approach you should take is to absorb the code from that addin into your own addin.
From the help for the pythonaddins module:
pythonaddinsmodule can only be used within a Python add-in. It cannot be used in stand-alone scripts or geoprocessing script tools.