Ubuntu 21.10 no driver loaded for Intel Intel TigerLake Iris Xe Graphics

I have basically the same problem as described here or here or here: the driver for my intel GPU is not loaded. As seen here

$ inxi -G
Graphics:
  Device-1: Intel TigerLake-LP GT2 [Iris Xe Graphics] driver: N/A 
  Device-2: NVIDIA TU117M [GeForce GTX 1650 Ti Mobile] driver: N/A 
  Display: x11 server: X.Org 1.20.13 driver: loaded: fbdev 
  unloaded: modesetting,vesa resolution: 1920x1200~77Hz 
  OpenGL: renderer: llvmpipe (LLVM 12.0.1 256 bits) v: 4.5 Mesa 21.2.2

or here (display unclaimed)

$ sudo lshw -c video
  *-display UNCLAIMED       
       description: VGA compatible controller
       product: TigerLake-LP GT2 [Iris Xe Graphics]
       vendor: Intel Corporation
       physical id: 2
       bus info: pci@0000:00:02.0
       version: 01
       width: 64 bits
       clock: 33MHz
       capabilities: pciexpress msi pm vga_controller bus_master cap_list
       configuration: latency=0
       resources: iomemory:600-5ff iomemory:400-3ff memory:601e000000-601effffff memory:4000000000-400fffffff ioport:4000(size=64) memory:c0000-dffff memory:4010000000-4016ffffff memory:4020000000-40ffffffff
  *-display UNCLAIMED
       description: 3D controller
       product: TU117M [GeForce GTX 1650 Ti Mobile]
       vendor: NVIDIA Corporation
       physical id: 0
       bus info: pci@0000:01:00.0
       version: a1
       width: 64 bits
       clock: 33MHz
       capabilities: pm msi pciexpress bus_master cap_list
       configuration: latency=0
       resources: iomemory:600-5ff iomemory:600-5ff memory:61000000-61ffffff memory:6020000000-602fffffff memory:6030000000-6031ffffff ioport:3000(size=128)

In all three mentioned cases above the problem was solved by

apt install linux-oem-20.04

since they where all running on Ubunut 20.04 and needed a newer kernel. Unfortunately, installing the OEM kernen did not do the trick for me. Any suggestions, how to fix this?

Some more information on my system, if more needed just tell me :)

$ uname -r 
5.13.0-25-generic

$$  lspci -nnk | grep -A3 "\[03..\]:" 
0000:00:02.0 VGA compatible controller [0300]: Intel Corporation TigerLake-LP GT2 [Iris Xe Graphics] [8086:9a49] (rev 01)
    DeviceName: Intel Tigerlake UHD Graphics
    Subsystem: Hewlett-Packard Company TigerLake GT2 [Iris Xe Graphics] [103c:8815]
    Kernel modules: i915
--
0000:01:00.0 3D controller [0302]: NVIDIA Corporation TU117M [GeForce GTX 1650 Ti Mobile] [10de:1f95] (rev a1)
    Subsystem: Hewlett-Packard Company TU117M [GeForce GTX 1650 Ti Mobile] [103c:8815]
    Kernel modules: nvidiafb, nouveau, nvidia_drm, nvidia

$ prime-select query
intel

$ glxinfo -B
name of display: :1
display: :1  screen: 0
direct rendering: Yes
Extended renderer info (GLX_MESA_query_renderer):
    Vendor: Mesa/X.org (0xffffffff)
    Device: llvmpipe (LLVM 12.0.1, 256 bits) (0xffffffff)
    Version: 21.2.2
    Accelerated: no
    Video memory: 15754MB
    Unified memory: no
    Preferred profile: core (0x1)
    Max core profile version: 4.5
    Max compat profile version: 3.1
    Max GLES1 profile version: 1.1
    Max GLES[23] profile version: 3.2
OpenGL vendor string: Mesa/X.org
OpenGL renderer string: llvmpipe (LLVM 12.0.1, 256 bits)
OpenGL core profile version string: 4.5 (Core Profile) Mesa 21.2.2
OpenGL core profile shading language version string: 4.50
OpenGL core profile context flags: (none)
OpenGL core profile profile mask: core profile

OpenGL version string: 3.1 Mesa 21.2.2
OpenGL shading language version string: 1.40
OpenGL context flags: (none)

OpenGL ES profile version string: OpenGL ES 3.2 Mesa 21.2.2
OpenGL ES profile shading language version string: OpenGL ES GLSL ES 3.20