The Dynamic Lattice Liquid (DLL) model has been implemented as a simulation algorithm for two-dimensional (2-D) polymeric systems. The model can work with density factor rho=1 and became a basis for a parallel algorithm, which takes into account coincidences of elementary molecular motion attempts resulting in local cooperative structural transformations. Systematic studies of athermal polymer systems on the two-dimensional triangular lattice are presented for various concentrations of linear polymers with respect to an explicitly considered solvent. Both static and dynamic properties are characterized. The simulation results reflect molecular packing and other properties of monomolecular polymer layers, which can be relevant to these obtained in some thin film formation techniques.