Methods to improve the efficiency and the memory requirement of LU factorization for complex symmetric system matrix

You probably want a factorization of the form $mathbf A = mathbf L mathbf D mathbf L^T$, it can certainly be applied to a complex symmetric $mathbf A$. LAPACK implements this factorization within [zsytrf] and provides a corresponding backsolution routine within [zsytrs]. There are sparse-direct versions of this algorithm too. PARDISO, TAUCS, and MyraMath all implement it (disclaimer: I wrote that last one).

EDIT1: Regarding the efficiency of backsolution, it's probably not great. Unlike LU [zgetrf] and Cholesky [zpotrf], the algorithm used by [zsytrf] technically does not deliver a triangular factor that is layout compatible with the triangular routines of the BLAS (eg [ztrsm]). Instead, it stores $mathbf L$ and $mathbf D$ interleaved as a bunch of 1x1 and 2x2 blocks (kinda arbitrarily, based on the pivoting decisions), which means the backsolution requires a similar sequence of rank-1 and rank-2 steps (this fussiness of the backsolution process is why LAPACK provides [zsytrs] to begin with). Unfortunately this is all BLAS1/BLAS2 level of performance. The algorithm to untangle $mathbf L$ into a BLAS3-compatible triangular matrix is tedious.

EDIT2: If your input is sparse, I'd just use a package that handles all this. Start with PARDISO, it's already present in MKL. It's probably not worth digging into any of the details.