This paper gives necessary and sufficient conditions for the existence of a positive definite or semidefinite solution to the symmetric recursive inverse eigenvalue problem (SRIEP) when the recursive matrix \(R\) is singular. Some computational issues are considered in order to simplify the formulae for proposed positive/semidefinite solutions of the SRIEP and to reduce the computational cost. Several numerical experiments are given. As the authors point out finding an (indefinite) solution to the SRIEP when \(R\) is singular is still an open problem. Also, the problems of the sensitivity analysis of the SRIEP or the error analysis for solving the overdetermined linear system are not considered.