Structural analysis is essential for understanding the characteristics of integro differential algebraic equations (IDAEs) before numerical analysis. The Σ-method, utilizing the signature matrix, effectively analyzes differential-algebraic equations (DAEs) and extends to IDAEs. Challenges arise when the signature matrix becomes undefined or overestimated due to integrated derivatives in IDAEs. Additionally, when a singular Jacobian matrix is yielded after applying the Σ-method, existing conversion methods may fail to ensure process termination. This paper addresses these issues by splitting an IDAE into two parts, redefining the signature matrix for each, and introducing a new degrees of freedom measure to ensure conversion method termination. Furthermore, a point-based on a detection method corrects signature matrix overestimation. Finally, an embedding method regularizes nonlinear IDAEs with singular Jacobian matrices. When coupled with the collocation method, it can effectively solve a general IDAE numerical