How do I reach balanced state in combat and how do I maintain it? How can I lose it?

a balanced state means your profile wins and losses are equal, you maintain it by keeping the won and lost fights of your profile in balance, you can see it by opening the round icon between the panels on the right side

it is possible to get won and lost fight without gaining profile wins as well, experiment to see it for yourself

Spoiler

if you are in an alliance there is a mechanic that can let you stay in perma balance as long as your honor value is positive, this means you will not gain profile wins or losses once you reached the perma balance

balance will give you a bonus on the stats you gain throught the fight

 

In addition to Mag's answer: there is also a wish in the wishshop that can help you with that.