Why can't I access any site while VPN is running?

Question

VPN

Good afternoon!
Good people, point the right way.
The problem is this: there is a computer that, when connected to the enterprise network through the cisco vpn client version 4+, suddenly stops accessing the Internet through the gateway of the user's home network router.

I ping the same ya.ru to which I get a response about the unavailability of this resource.

I decided to check the routing table, but found nothing interesting there.
OS Windows, the picture in the IPv4 table is approximately * this:
0.0.0.0 0.0.0.0 192.168.0.1 metric 51 - user router
10.10.0.0 255.255.0.0 10.10.10.1 metric 2 - enterprise mesh
all other routes have a lower priority.
Those. everything seems to be ok. I start the tracer up to 8.8.8.8 - and I see that the first hop is the user's router, so everything is ok.

I assume that the problem may be in the DNS.
I turn to Google dns with a request to give ip for ya.ru.
I ping ya.ru by ip and really - it responds. I let the tracer catch up - the packets fall where they need to.
I prescribe static DNS to user 8.8.8.8 on all interfaces.
I open the browser and try to go to ya.ru - the page is not available "dns probe finished bad config".

Again I ping Yandex by name - again I get a message about the unavailability of the resource.

I close the browser and open cmd, type in the following commands:
ipconfig/flushdns
netsh int ip reset
netsh winsock reset.
shutdown -r -t 0
I open the browser again - nothing has changed.
Again I ping Yandex by name - again an error.

There was another attempt to prescribe route add -p 0.0.0.0 mask 0.0.0.0 192.168.0.1 metric 1, but this did not help.
I also tried to register a Google server in the dns settings on the router itself - also without success.
I also tried to simply reinstall the client, tried to reinstall to an older and more stable version, also tried a major reinstallation with cleaning the registries and folders with configurations from ProgramData and AppData, but something somehow didn’t work either.
I also tried to force split-routing in the settings of the client itself, allowing most of the traffic to go to the user's router - also no effect.

Has anyone experienced something similar in their practice? What can you recommend, comrades?