A line system (1) for fluids having volatile components, which comprises at least one fluid line (2,3,4), each having a separate inlet (10) and outlet (11), the fluid lines (2,3,4) being enclosed by a first line wall (12) and extending jointly over a section (A) of the line system (1) close-fitting to one another, but being separated from one another at least by the first line wall (12) over the entire length of the fluid lines (2,3,4). The line system (1) is implemented as a plastic hollow profile at least over the section (A) and comprises a further fluid line (8), which is enclosed by a second line wall (13), and comprises one or more line chambers (8'',8'''), which are at least partially separated from one another by support webs (7), whose totality at least essentially encloses the fluid lines (2,3,4). In addition, the first and second line walls (12,13) and the support webs (7) of the plastic hollow profile being implemented from the same polymer material. The line system (1) according to the present invention is characterized in that the further fluid line (8) is implemented as a scavenging line (8) having scavenging chambers (8'',8''') and is connected to a scavenging device, which is implemented for flushing or suctioning the scavenging line (8) and therefore for conveying away and collecting permeates originating from the fluid lines.