Kimi no Kakera

From Mangascreener:

Icoro is the princess of a country of eternal night. Her kingdom is surrounded on all sides by huge mountain walls. Snow falls constantly, yet without the warmth of the sun's daylight, it will never melt, only pile up higher and higher. Her people are destined to die out. While she is a member of the ruling royal family, her parents have both disappeared, leaving her alone with her blind younger brother, and crafty politicians hold all the power behind the scenes, leaving her sovereign in name only. Her life has been an endless cycle of hardship, and she has never smiled, not once. In fact, she is missing that part of all humans that knows happiness.

Enter Shiro, a mysterious young boy whose identity is an enigma, even to himself. He is able to forget all but the most vital of information within seconds. All he knows is that he is looking for something called a "Hitogata." After Icoro and Shiro's chance meeting, they end up on a desperate search for the sun that will melt the snow and bring happiness to the kingdom once more.

Will they ever escape the grasp of the corrupt authorities? Will they even recognize what they're looking for, if they come across it? Will this "Hitogata" be the key to finding a sun to save their home? The answers lie ahead.