Isochrones of systems with a small parameter that are independent of it

We consider an autonomous system on the plane involving a parameter λ, commonly referred to as a small parameter, and with a singular point (either a center or a focus) at the origin. It is assumed that the system is isochronous at this singular point. We also assume that its isochrones – defined as the pre-images of radial lines under the transformation from the system under consideration to a system with uniform isochronicity – are independent of the parameter λ. Thus, in this paper, the parameter λ may take on any real value and, in that sense, is not actually small. We construct systems with quadratic nonlinearities in the right-hand side that satisfy these requirements and determine their isochrones. Phase portraits of the resulting isochronous foci and their isochrones are also constructed.