Let  (S) $Y' = A(x)Y$  be a  system of first order linear differential equations with rational functions coefficients.  A  singular point  $x_0$  of (S) is  called an apparent singularity if  there is a basis of formal solutions of (S)  which are holomorphic in a neighborhood  of $x_0$.   In this talk we shall present a new  algorithm which, given a system of the form (S),  detects  apparent singularities  and constructs an equivalent system (S') with rational coefficients , such that every singularity of  (S') is a singularity of ( S) that is not apparent.    Our method can, in particular, be applied  to the companion system  of any  linear differential  equation with arbitrary order $n$ . We  thus have an alternative  method  to the standard methods  for removing apparent singularities of linear differential operators.  We shall compare our method to the one designed for operators and we shall show some  applications and examples of computation.