This circuit based on some circuits found on the Internet.
It was simplified and modeled in Multisim and then checked. It works fine!
This circuit contains 2 white LEDs, solar battery (2 volts) and rechargeable battery (1.2 volts)
Also you may remove 100k resistor, and resistors 10k and 6.8k may be changed to near values.
Transistors also may be changed to some NPN.
L2 may be from 300 to 1000 uH
PCB in Sprint Layout
PCB made by LUT 🙂
Applying mask, but with some slide 
It works! 
Some investigation in oscilloscope…
PCB with elements
