  Integro-differential-algebraic equations (IDAE)s are widely used in
applications of engineering and analysis. When there are hidden constraints in
an IDAE, structural analysis is necessary. But if derivatives of dependent
variables appear in their integrals, the existing definition of the signature
matrix for an IDAE cannot be satisfied. Moreover, if an IDAE has a singular
Jacobian matrix after structural analysis by the Sigma-method, improved
structural analysis methods are proposed to regularize it. However, the optimal
value of an IDAE may be negative which can not ensure the termination of the
regularization. Furthermore, overestimation of the signature matrix may also
lead to failure of its structural analysis.
  In this paper, firstly, we redefine the signature matrix and introduce a
definition of the degree of freedom for IDAEs. Thus, the termination of
improved structural analysis methods can be guaranteed. Secondly, the detection
method by points is proposed to deal with the problem of overestimation of
signature matrix. Thirdly, the embedding method has proved to suitable for
structural unamenable IDAEs, including those types that arise from symbolic
cancellation and numerical degeneration. Finally, the global numerical method
is applied to an example of two-stage drive system which can help to find all
solutions for IDAEs by witness points. Hopefully, through the example of
pendulum curtain, the approach for IDAEs proposed in this paper can be applied
to integro-partial-differential-algebraic equations (IPDAE)s.
