using onboard grafic

I'll add to Richie's comments the point that of the two currently active Einstein GPU applications, the Gamma-Ray Pulsar one is far, far less demanding of support from the host system CPU and motherboard.  People run AMD RX570 graphics cards at nearly their full capability on systems with old and modest CPUs.  However the current Gravity-Wave application makes rather heavy use of the CPU, so a modest CPU capability system will obtain a smaller fraction of possible output from the GPU.

It could prevent the most recent GPUs (hundreds of euros) from working but that's because they require a newer motherboard (must have UEFI, BIOS not compatible anymore). But older and cheaper "PCI-e 3.0 x16" GPUs are compatible with older motherboards that have BIOS and PCI-e 2.0 x16. My motherboard is Asus P5Q Pro (PCI-e 2.0 x16) from 2009 (your Asus A3N WS is from 2009 too). I don't have any problems running Nvidia GTX 1060 on this but I'm pretty sure that GTX 10xx-series is the limit for this and RTX 20xx-series cards woudn't work at all.

Perhaps similar sort of thing here... talking about some Adaptec RAID card:

https://ask.adaptec.com/app/answers/detail/a_id/16991/~/why-does-the-adaptec-pci-express-card-not-work-in-a-pcie-x16-slot%3F

"The problem here is that the Adaptec BIOS code will be copied into the different memory area which is reserved for VGA cards and this corrupts the video output. Hence there is no video output."

You need to

1. make sure that correct GPU drivers for that GPU are installed in Linux.

2. visit https://einsteinathome.org/account/prefs/project and choose a 'Preference set' and set that same 'Location' for your host (accessible through your computers page at this Einstein site). Then check the box 'Use NVIDIA GPU' or 'Use AMD GPU', which ever card you had. And choose the correct GPU applications that you like to run.

3. Also, allow Boinc to use your GPU. There are a couple of simple things to check on the Boinc Manager:

Boinc Manager - Activity menu ... Run GPU (Always / Based on preferences / Suspend GPU)

Boinc Manager - Options - Computation preferences... - Computing tab - When to suspend

(those options can alternatively be accessed via command line and configuration file if Boinc Manager not preferred).

What PSU do you have by the way (make, model)? Well, let's assume it's not extremely old and can provide enough power for a GPU that has max one 6-pin power connector. I would say... for FGRPB1G tasks:

From Nvidia: GTX 750, GTX 750 Ti, GTX 950, GTX 960

From AMD:

GCN 1.0 cards (these are only for FGRPB1G, not compatible with Gravitational Wave GPU tasks)

GCN 2.0 cards (filtered list, only models that have been available with a single 6-pin power connector)

There would be more GPU models on the lists if 2 x 6-pin or 8-pin power connectors were allowed. But that means the PSU would have to tolerate them. Those older AMD cards could then easily mean also more noise, heat and electricy consumption. An example of such card... maybe Radeon R9 270X. It could still crunch FGRPB1G tasks, but takes some juice from the socket (TDP 180W). I don't know how easy it would be to throttle it down under Linux. That GPU model also had 'GCN 1.0' architecture and wouldn't be able to run GW GPU tasks at all.

Since you could use a very wide range of relatively modern GPUs (PCI-e 2.0 is NOT a problem) the more important thing to think about is the quality of your PSU and its 12V power spec.

Since you also are running an old (and slow) CPU, and since you imply that you have a tight budget and therefore that you're interested in keeping costs down, here are a couple of suggestions for you to consider.

#1 - A second hand RX 460 or RX 560 - the type that doesn't need an external PCI-e power connector, but is able to run with what it can draw from the x16 slot (the spec for that is 60W).  I'm running quite a few of these using 300W PSUs.  The PSUs are quite old but they are of the modern 12V 'heavy' design and are 80+ efficient.  Their rating is 270W on the 12V rail.

An example of mine is a machine with a Pentium G640 dual core CPU.  It had a long term stable RAC of 240-250K.  With the change to the LATeah3001 and above tasks, the RAC is now 330K and wont be stable for another week or two.  It should get to around 360K if the current type of tasks continues.  It crunches 2 tasks in ~27.5 mins.

#2 - A second hand (or new if you can find one at the right price) RX 570.  The 8GB variety would be preferable if you ever want to run multiple GW GPU tasks - although you would need to do a full upgrade to a more modern board, RAM and CPU to run those.  The 4GB version is quite OK with the current host for GRP GPU tasks since there is no real gain in running more than 2x and only minimal CPU support is required.

An example of mine is a machine I built in 2008 based on a Q6600 core 2 quad CPU.  Its x16 slot is PCI-e 1.x - not even 2.0.  It was upgraded around 3 years ago with a 4GB RX 570.  It still has the original board and 2x2GB DDR2 RAM.  I power the system with a 450W 80+ efficient PSU that is rated for 410W on the 12V rail.  A single 8pin PCI-e power connector is required.  It had a long term stable RAC of ~500K and with the new tasks is now at ~700K and still rising.  I expect it to get to around 730-740K RAC.  It crunches 2 GRP tasks in ~13.5 mins.