Abstract The purpose of this work is to obtain sufficient conditions of a solution existence and uniqueness for a class of inverse problems for linear evolution equations with a degenerate operator at the derivative and with an unknown element in the right-hand side of the equation, which depends on the time variable. The overdetermination condition is given on the kernel of the operator at the derivative, the initial condition have the Cauchy form or the Showalter–Sidorov form. The obtained abstract results are applied to the investigation of linear inverse problems for the Sobolev system of equations and for the linearized Oskolkov system with overdetermination on the pressure gradient function.