Just to clarify, I don't think your issue is related to the Data ONTAP Toolkit or really even PowerShell or NetApp, so you might get better help elsewhere... but I'll try anyway!
The thing I noticed immediately is that you are using the "nt authority\system" account which I believe is restricted from accessing network resources. I'm not sure what you are trying to accomplish, but you probably should try to use a domain account. I'm guessing you are trying to access the shares from a service or system process. Hopefully you can assign the user that service runs under to a regular domain account. It might still be necessary to map the drive with "net use" before accessing it since services won't necessarily trigger the standard windows login processing that maps drives.
i think also thats not a problem from our NetApp, i'm just the stupid Netapp Admin who try to help. I told him already to change to a Service Account. They use this nt authority\system to distribute software over other workstations.
They use net use as first step but when they want access the share they get a strange error message, like you can see in the screenshots. Maybe someone else tried this and can help this guy from my company.
Else i will repeat "use a service account" and will close the discussion from my side.
If the NT workstation is joined to the same domain as the filer, then the system account equates to the computer account for the workstation in the domain. Suppose I have a domain, mydomain, and a workstation joined to the domain, myworkstation. Add mydomain\myworkstation$ to the permissions on the share and volume. If the filer and/or workstation are not domain joined, you're out of luck.