  We explore the physical properties of a unified microscopic theory for the
coexistence of superconductivity and charge density waves in two-dimensional
transition metal dichalcogenides. In the case of particle-hole symmetry the
elementary particles are Dirac fermions at the nodes of the charge density wave
gap. When particle-hole symmetry is broken electron (hole) pockets are formed
around the Fermi surface. The superconducting ground state emerges from the
pairing of nodal quasi-particles mediated by acoustic phonons via a
piezoelectric coupling. We calculate several properties in the s-wave
superconducting phase, including specific heat, ultra-sound absorption, nuclear
magnetic relaxation, thermal, and optical conductivities. In the case with
particle-hole symmetry, the specific heat jump at the transition deviates
strongly from ordinary superconductors. The nuclear magnetic response shows an
anomalous anisotropy due to the broken time-reversal symmetry of the
superconducting gap, induced by the triple charge density wave state. The loss
of lattice inversion symmetry in the charge density wave phase leads to
anomalous coherence factors in the optical conductivity and to the appearance
of an absorption edge at the optical gap energy. Furthermore, optical and
thermal conductivities display anomalous peaks in the infrared when
particle-hole symmetry is broken.
