(4+1) and (3+1+1) reductions of the geodetic equations in 5D theory with a scalar field and out of 5D optics lead to the new concept of the Lorentz-type 5D mass of the particle. Due to the imposition of the x⁵-cylindricity condition, one can obtain an integral of the 5D particle’s motion along the x⁵, that gives one an electric charge of it. Thus, one can obtain an exact expression for 5D mass, that depends on the scalar field through an electric charge, so maybe there is no need in a scalar charge in the Nature. One can compactly express 5D mass through the mass angle and additionally to hypothesize about the possible complex structure of it. It leads one to the deeper understanding of the quantum properties of the matter. All of these new conceptions turns one also to the idea, that 5D mass may contribute to the recently discovered in the Universe dark matter and dark energy and also to be one of the possible reasons of the Universe’s expansion. The next suggestive result in 5D theory is connected with the (4+1) reduction of the 5D Ricci identities. It leads one to the certain connections between the 4D physico-geometrical values and permits one to obtain the first pair of the Maxwell equations with the nonzero soliton-type r.h.s. and establishes the connection with the second pair of them. It leads one to the idea about the magnetic monopoles’ existence in the early Universe. It is shown, that this nonzero r.h.s. vanishes together with the imposition of the x⁵-cylindricity condition. This process permits one to hypothesize soundly about the existence of the topological second-order transition of the Universe’s matter, which leads it to the superfluid state and maybe accelerates its expansion.