In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm-Liouville operator L generated in L 2 (R+, S) by the differential expression ` (y) = −y 00 + Q (x) y , x ∈ R+ : [0, ∞), and the boundary condition y0(0) − β0 + β1λ + β2λ 2 y (0) = 0 where Q is a non-selfadjoint matrix valued function. Also using the uniqueness theorem of analytic functions we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities