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After you add the .exe in the UWP project, you need to declare it in the manifest and also add the 'runFullTrust' capability in it. You can follow the steps below. About more details, you can refer to this similar thread.
1.Add "Windows Desktop Extensions" reference in the project. Right click the References-> Add Reference->Universal Windows->Extensions-> add Windows Desktop Extensions".
2.Open manifest, add Extensions tag and runFullTrust capability. Suppose you put the .exe in the assets directory. About the full content of manifest, you can view it from here.
3.When you click the button, use the following code to start your .exe.
private async void Button_Click(object sender, RoutedEventArgs e)
{
await FullTrustProcessLauncher.LaunchFullTrustProcessForCurrentAppAsync();
}