Thanks for understanding me and I would say increase the number of batteries. Will a 9V rechargeable battery be okay? Also If increasing the number of solar cell and remain the 2 x 1.2V battery, how many pieces do I need? I am opening to any options except the 3 main points which I need to achieve.I'll try one more time to understand what your constraints are.
If you are limited to six solar cells that produce a maximum of 2V at 150mA each, and you want to recharge the batteries using only those cells, then you must not draw more power from the batteries than the cells can produce. In the best case (bright sun, best angle), those six cells will produce 1.8 watts. Realistically, they probably will not average more than one watt output during daylight hours. So, you have only one watt that you can draw from your batteries if you expect the solar cells to recharge the batteries.
Next, if you are limited to two 1.2V batteries, then you can connect them either in parallel, in which case you have 1.2V, or in series, in which case you have 2.4V. Thus, the only LEDs which you could light would be those with less than 2.4V drop, i.e., the red (1.8 - 2.3V) and the IR (1.5 - 1.5V), and that would be with the batteries in series. Without some voltage boost circuit, it would be impossible to light the blue or the UV.
So, you don't have enough voltage from the batteries to light the LEDs, and you don't have enough current from the solar cells to recharge the batteries. Therefore, you must have more batteries and more solar cells in order to light all the LEDs you have listed to full brightness.
What options will you consider? More batteries, more solar cells, or fewer LEDs?