One Servlet as Web Servlet, REST Servlet and WebSocket Servlet

Is there any servlet in JEE which understands all three APIs?
The problem is that there is no servlet which handles requests from all these three under one Security Context. At the same time the Model (in MVC) should appear on client side and server should become stateless.
This means the web (xhtml), rest API and websocket API in one servlet and understands same Security Context if the request is coming from the page or REST or Websocket. The servlet should understand the CDI context as well.
I do not want to use Angular and I think it is the time when JEE discontinues JSF and creates something like Java Client Faces and the backend will be a servlet which understands HTML5 + REST + WebSocket.
The Web Servlet should resolve static expressions, then the same servlet should understand AJAX and REST requests under the same Security Context. Same with Websockets. The REST calls on server side must understand the JSESSIONID, roles, etc.
The xhtml page should use expressions. Servlet should resolve them to static strings which prevents from producing too many async calls to the server.