In this paper,for the regularized Hermitian and skew-Hermitian splitting (RHSS) preconditioner introduced by Bai and Benzi (BIT Numer Math 57:287-311,2017) for the solution of saddle-point linear systems,we analyze the spectral properties of the preconditioned matrix when the regularization matrix is a special Hermitian positive semidefinite matrix which depends on certain parameters.We accurately describe the numbers of eigenvalues clustered at (0,0) and (2,0),if the iteration parameter is close to 0.An estimate about the condition number of the corresponding eigenvector matrix,which partly determines the convergence rate of the RHSS-preconditioned Krylov subspace method,is also studied in this work.