Is money earned permissible or impermissible in the case of working for an institution owned by one's brother and receiving a salary from the state, with no specific tasks?

If the entity from which you receive a salary knows that you do not have a specific job and considers you one of its employees, and it has agreed to pay the salary, then there is no sin upon you in taking it. However, if it is merely a ploy on your part to take the salary without informing them of the reality of the situation, then it is not permissible for you to take it, because the sanctity of public funds is greater. The Prophet, peace and blessings be upon him, warned against taking public funds unjustly, explaining that whoever does so will meet Allah Almighty on the Day of Resurrection carrying it, and that he will have the Fire on the Day of Resurrection.