I am not sure what is the correct topic for my question; 1. Is there

Yes there is, although you'd have to leverage the Web Service API component in order to have the external system manage groups within Appian.

There is a component in the Shared Component section that lets you do LDAP User Sync. However it does not sync groups. We generally don't recommend managing groups external to Appian because groups in Appian should be used for Appian-specific object security and visibility. However, you could build process models that create/delete/update groups and expose them as web services for invocation by an external system.

I am revisiting this topic. Let me change my questions. We are using LDAP to sync users(Authticate). WE are using Appian admin tool to manage groups (authorize). My questions is, if some how my system crashed and I need to redeploy to a new hardware, is there a way I can /backup/reload the users to group?

Hopefully this answers your question. I would create an application that would contain all my groups. Export that application. Then in my new environment, I would run the LDAP sync to create the users and then I would import that group application into my new environment. Not sure if this is still true, but during the group application import the newly created groups would populate with the member structure of your previous environment.