Integrating Jot with a custom user system might be done as follows:
1) tell Jot to use explicit field for username
[[Jot? ... &customfields=`name` ...]]
2) do some changes in Jot’s form by inserting hidden field to keep username
<input type="hidden" name="name" value="[[get_user_name]]" ... />
3) code "get_user_name" snippet -- its implementation entirely depends on technical details of your custom user system, anyway it should be something like
if (<user-logged-into-custom-system>) {
return <logged-user-name>;
} else {
return <anonymous-name>;
}
And the most important, the above is merely an outline, not a complete solution.