A customer service ticket (issue) that is created with a Jira Service Management form (project) can be used to trigger a Next Matter instance. Here is how it works in three simple steps.
Create the automation rule to trigger the Next Matter process
Hand over additional data from Jira to Next Matter by completing the first step (Optional)
Send back data along the way (Optional)
Step 1: Create the automation rule to trigger the Next Matter process
Before you begin, please check that nextmatter.com is allowlisted in Jira as a destination.
Go to the right project in Jira Service Desk (i.e. the form that you want to connect with Next Matter)
Navigate to Project Settings --> Automation
Create a Custom Rule and use "issue created" as the "When" Trigger
Add a "Then" Action and Use "Send web request"
As URL: Select the instance endpoint of Next Matter
As Headers add:
Content-Type set application/json
Authorization: Api-Key YOURKEY [If you don't have an API-Key yet, please reach out to Next Matter customer success team]
Method: Post
Webhook Body: Custom Data
{"name": "{{issue.key}}", "process": "https://core.nextmatter.com/api/processes/[IDOFTHEPROCESS]/" }
The ID of the process can be found in the url if you navigate to the respective process in your Next Matter Workspace
In this example, the key of the newly created jira issue is used as the name of the launched Next Matter process instance. Configured example looks similar to the below.

If you already publish the rule now the following will happen: Everytime that a Jira issue for this project is created, it will launch the specified Next Matter process and use the issue-key as the name of the instance.
Step 2: Hand over additional data from Jira to Next Matter by completing the first step (Optional)
You might want to hand over additional data from Jira to Next Matter by completing the first step of the launched Next Matter process. This can be done by adding an "And:" action to complete a subsequent step.
Add "And" with another web request
As Webhook URL add
https://core.nextmatter.com/api/instances/{{webhookResponse.body.id}}/complete_step/
Headers and Method are same as in the first request
As custom body, you can map jira issue information to corresponding Next Matter actions. Next Matter actions can be identified with the ID that is displayed in the editor.
The id of the step itself (step_id) is shown in the URL
Sample Configuration Screenshot

Done. Test it out by creating a new issue, for example by filling out the public service request form. The result should be a newly created instance and a filled out first step.
Screenshot of the sample result:

Step 3: Send back data to the issue along the way
With the issue key of the Jira issue known, you can make updates, for example a comment that is sent to the user that raised the issue originally with an integration step. This is described in this article.
Further Resources
Using Jira Webhooks: https://confluence.atlassian.com/servicemanagementserver/send-alerts-with-jira-service-management-webhooks-968670084.html
Using Webhook Response Body: https://confluence.atlassian.com/automation074/send-web-request-action-now-provides-response-data-1141481224.html
List of available smart values to populate Next Matter steps: https://confluence.atlassian.com/automation/smart-values-993924860.html
Configuring the allowlist of Jira: https://confluence.atlassian.com/adminjiraserver/configuring-the-allowlist-1014667309.html