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1. Conditional symmetries and conditional constants of motion for dynamical systems

Giuseppe Gaeta.

Conditional symmetries were introduced by Levi and Winternitz in their 1989 seminal paper to deal with nonlinear PDEs. Here we discuss their application in the framework of ODEs, and more specifically Dynamical Systems; it turns out they are closely related to two established -- albeit maybe less widely known -- concepts, i.e. orbital symmetries and configurational invariants. The paper is devoted to studying the interplay of these notions, and their application in the study of Dynamical Systems, with special attention to invariant manifolds of these.

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2. The N-species integrable Volterra system as a maximally superintegrable Hamiltonian system

Orlando Ragnisco ; Federico Zullo.

The results presented in this paper are a natural development of those described in the paper {\it The Volterra Integrable case. Novel analytical and numerical results} (OCNMP Vol.4 (2024) pp 188-211), where the authors reconsidered the integrable case of the Hamiltonian $N$-species Lotka-Volterra system, introduced by Vito Volterra in 1937. There, an alternative approach for constructing the integrals of motion has been proposed, and compared with the old Volterra approach. Here we go beyond, and show that in fact the model introduced by Volterra and studied by us is not just integrable, but is maximally superintegrable and reducible to a system with only one degree of freedom regardless the number of species considered. We present both analytical and numerical results.

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