Doesn't inject EJB in different servlets

Question

I have at least 2 Servlets and try to share Statefull bean between them. But, unfortunatelly, I can't because it inits for each Servlet.

@WebServlet(name = "FirstServlet", urlPatterns = {"/first/*"})
public class FirstServlet extends BaseServlet {

    @EJB
    private IUserSession userSession;

    protected void doGet(HttpServletRequest request, HttpServletResponse response)
        throws ServletException, IOException {

          userSession.setUserName("my name");
         // todo something and put breakpoint here
    }
}


@WebServlet(name = "SecondServlet", urlPatterns = {"/second/*"})
public class SecondServletextends BaseServlet {

    @EJB
    private IUserSession userSession;

    protected void doGet(HttpServletRequest request, HttpServletResponse response)
        throws ServletException, IOException {
          
         // todo something and put breakpoint here
    }
}


public interface IUserSession {

    void initItems();
    void setUserName(String userName)
}

@Stateful(name = "userSession")
public class UserSession implements IUserSession {

    private String userName;

    @PostConstruct
    private void init() {
        initItems();
    }

    public void initItems(){
      // todo something
    }

    public void setUserName(String userName){
      this.userName = userName;
    }
}

So, while I am debugging, I can see, that before call doGet FirstServlet we go in to UserSession.init() method because it is PostConstruct. After we go into FirstServlet.doGet(). When I do the second request in the same session via browser and go to the SecondServlet I again go into UserSession.init() and after that, of course SecondServlet.userSession.userName return null.

How to share UserSession inside Application? I want to construct some... SessionBean for user information. I know, that I can put it inside request like

request.getSession().setAttribute(USER_SESSION_KEY, userSession)

but I want to use Enterprise mechanism. I am sure that I lost something easy but important.