Field theoretical Lie symmetry analysis: The Möbius group, exact solutions of conformal autonomous systems, and predictive model-building

• We explore classes of first order ODEs that admit Möbius transformations.
• We investigate autonomous systems that admit conformal transformations.
• We suggest a connection between Lie symmetries and phenomenology.

We study single and coupled first-order differential equations (ODEs) that admit symmetries with tangent vector fields, which satisfy the N-dimensional Cauchy–Riemann equations. In the two-dimensional case, classes of first-order ODEs which are invariant under Möbius transformations are explored. In the N dimensional case we outline a symmetry analysis method for constructing exact solutions for conformal autonomous systems. A very important aspect of this work is that we propose to extend the traditional technical usage of Lie groups to one that could provide testable predictions and guidelines for model-building and model-validation. The Lie symmetries in this paper are constrained and classified by field theoretical considerations and their phenomenological implications. Our results indicate that conformal transformations are appropriate for elucidating a variety of linear and nonlinear systems which could be used for, or inspire, future applications. The presentation is pragmatic and it is addressed to a wide audience.