In Webster dictionary the word Integrity -- Firm adherence to a code, especially moral or artistic values; incorruptibility has been the word that most searched in 2005. If it would tell you something it would tell you how people are eager to have integrity everywhere in public and private lives. To establish integrity in a society you have to find the ways that allow you to do so:
1- Group act: Integrity as other moral values or as I say it is the abbreviation of moral values have to be a group act.
2- The row models: In order to build a hierarchy of a community that have integrity you have to Put on the top the best of us regardless of their religion, race , color and so forth.
3- Democracy: the first enemy to corruption is democracy. That is no wonder that most of the non-democratic regimes are corrupt.
4- Collective consciousness: People in closed doors tend to be corrupt while they are likely more moralistic when they feel they are in the open air. So in every place people have to feel the public eye. If they are religious they would have double dose since they will feel before that God's eye.
5- Free responsible and incorruptible media: this will make it possible to people to keep an eye on politicians, government officials and all areas that may have the potential to corrupt.
6- Take away the avenues of corruption: like extreme poverty, closed medical records, motor vehicle accidents insurance with minor injuries used by corrupt people to drain the system, frivolous law suits and so forth.
7- Teaching morality and ethics even separate from religion in the education system.
8- Cut down or eliminate social causes that lead to dysfunctional families: alcoholism, drugs, high divorce rate, sexual abuse, child abuse,
9- Create hopeful societies in early life of individual through health and long term security.
10- Teach honesty and transparency as way of life instead of secrecy that create and encourage corruption. It is understandable that people do not or should not know every thing about you but there is things that they have to know like a patient who knows his medical record.
11- Teach forgiveness that allow people to admit their mistakes and feel no guilt and fear so they learn to lie and corrupt.
12- Discourage the pick on, mean-spiritism and litigation that disintegrate the communities by teaching kindness and morals.
As you see integrity is a code of moral ethics and to integrate it we have to get all of us have a better integrity. As a societies we will feel we are united by the moral code of ethics or righteousness if we to speak religion.