Hi Vladislav,
Is your GW system and BE system are different ?
Check your RFC connection you used with the service system alias, and do remote logon , then check your inbox again to see if any task exist,
sometimes the RFC destination is mapped to a specific user and this user might not have any tasks in his inbox.
Assuming you have tasks on your BE system, then one of the two happens
1. it's a task from an activity step , and hence need to be properly externalized
2. it's a task from a user decision step and someone enable the filter flag (you can disable it or do the same as follow)
To mark the tasks to allow externalize it (also when filter is enable):
On your BE system open Transaction SPRO
go to SAP Customizing Implementation Guide-> SAP NetWeaver -> Gateway Service Enablement -> Content -> Workflow Settings -> Maintain Task Names and Decision Options
A cluster view will open up and you need to fill it (workflow id, step id , short description)
For example :
Workflow ID = WS90000004, Step ID = WS90000004, Description= Leave Approval
Then click enter, select the line and select the decision Keys on the left side
then add new decision keys and text according to your workflow step outcomes,
save it and test again
BR
Roy