I am implementing LIve Agent chat. What I noticed is that Case is created and Contact is created but Case and Contact are not linked. Both are linked to chat transcript., But when I naveigate to Case detail there is no contact linked.

We have Person Account Enabled in Org.

When I implement  following trailhead in develiper org where is no Person Account enabled then I can see Case and Contact linked in Case detail page.
Link to trailhead
https://trailhead.salesforce.com/en/search?keywords=web%20chat%20basics

scenario :   Case is created in Salesforce,  Immediatly this case details be moved to third patry systems ( .net system, java sysstem,.......)
how can i do this ?
example : 
case # 12345 created in salesforce, immediatly case reflected in third party systems like java ,.net or some other systems.
Due to project cost, not going for opiton use third party integration tools like Dell boomi......or any other tools.

please provide any ideas on this requirement 
Thanks