Yes, you can set up Jira to automatically send email notifications to a particular person when tickets are not assigned and/or when due dates are not available for tickets.
This can be done by creating a Jira automation rule.
Here are the steps to create an automation rule for this:
Go to the "Automation" section of your Jira project.
Click on the "Create rule" button.
Choose the "Scheduled" trigger and set the frequency to your desired interval (e.g. every hour).
Add the "Find issues" action and set the JQL query to find issues that are not assigned and/or have no due date.
Add the "Send email" action and set the recipient to the particular person you want to notify.
Customize the email message as desired and save the rule.
Once the automation rule is set up, it will run at the specified interval and send an email notification to the designated person if there are any issues that match the JQL query.
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