Google Play Services Sync problem on Android 2.3.6
After looking through verbose log of my mobile phone I noticed that Google Play Services (GMS) can't find class "android.os.UserManager" and it has another dead codes connected with Google Play Services (gms). 



And this blocks of errors apear over and over again so Google Play Services restarts them self. Class "android.os.UserManager" require API level 17; my phone has API level 10 so I decided to install the older versions of Goolge Play Services. 
After that I find out that problems appear with versions of Goolge Play Services greater then 8.3.0. With 8.3.0 and below there is no such problem. Google+ works fine with this version of GMS.

I install all programs that I need using Play Market and after that I delete all data and updates of Play Market to prevent updating Goolge Play Services (I find out that Play Market respones for updating Goolge Play Services).