The inverse problem for linear operator-differential equation of the first order with degenerate operator near derivative, with unknown function in the right-hand side of the equation and generally speaking with integral condition of overdetermination are investigated. It is shown that under the assumptions on the operator in equations that guarantee the existence of analytic resolving semigroup of homogeneous equation the satisfying of some concordance condition for initial and final datas is necessary for the solvability of the problem. But if one put initial data only on the image of the resolving semigroup then the conditions of well-posedness of the equations can be found.