Summary: A Lorentzian paracontact manifold \((M,\varphi,\xi,\eta, g)\) is called LP-Sasakian with coefficient \(\alpha\) if there exists a smooth function \(\alpha\) on \(M\) such that \[ (\nabla_Z\Phi) (X,Y)= \alpha \bigl(g (\varphi X,\varphi Z)\eta(Y)+g(\varphi Y,\varphi Z)\eta(X) \bigr), \quad\Phi(X,Y)= \frac{1}{\alpha} (\nabla_X\eta)Y, \] for any vector fields \(X,Y,Z,\Phi\) denoting the tensor field such that \(\Phi(X,Y)= g(X, \varphi Y)\). With any LP-Sasakian manifold \((M, \varphi,\xi,\eta, g)\) are associated three distributions \(D,D^+,D^-\), where \[ D=\text{Ker} \,\eta=D^+\oplus D^-,\;D^+=\{X\in TM;\varphi X=X\},\;D^-=\{X\in TM :\varphi X=-X\}. \] The authors prove that \(D,D^+,D^-\) are integrable and each maximal integral submanifold of \(D\) is locally a Riemannian product \(N^+\times N^-\), \(N^+,N^-\) respectively denoting integral submanifolds of \(D^+,D^-\). Furthermore, any LP-Sasakian manifold, as well as each integral submanifold of \(D\), is endowed with a CR-structure.