The Amazon basin has the largest number of fish in the world, and among the most common fishes of the Neotropical region, the threespot (Leporinus friderici) is cited, which in relation to its microparasitic fauna, has described only 1 species of the genus Henneguya, Henneguya friderici. The Myxozoa class is considered an obligate parasite, being morphologically characterized by spores formed by valves connected by a suture line. This study describes a new species of Henneguya sp. in the Amazon region for L. friderici. This parasite was found in the host's pyloric caeca and caudal kidney, with mature spores with a total spore length of 38.4 ± 2.5 (35.9–40.9) μm; the spore body 14.4 ± 1.1 (13.3–15.5) μm and 7.3 ± 0.6 (6.7–7.9) μm wide. Regarding its 2 polar capsules, they had a length of 5.1 ± 0.4 (4.7–5.5) μm and a width of 2.0 ± 0.1 (1.9–2.1) μm in the same pear-shaped, and each polar capsule contained 9–11 turns. Morphological and phylogenetic analyses denote that this is a new species of the genus Henneguya.

Let $\sigma _q \,:\,{{\mathbb{R}}^q} \to{\textbf{S}}^q\setminus N_q$ be the inverse of the stereographic projection with center the north pole $N_q$. Let $W_i$ be a closed subset of ${\mathbb{R}}^{q_i}$, for $i=1,2$. Let $\Phi \,:\,W_1 \to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism $\sigma _{q_2}\circ \Phi \circ \sigma _{q_1}^{-1}$ is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at $N_{q_1}$ with value $N_{q_2}$ whenever $W_1$ is unbounded.

As two straightforward applications in the polynomially bounded o-minimal context over the real numbers, we obtain for free a version at infinity of: (1) Sampaio’s tangent cone result and (2) links preserving re-parametrization of definable bi-Lipschitz homeomorphisms of Valette.