For the May–Leonard asymmetric system, which is a quadratic system of the Lotka–Volterra type depending on six parameters, we first look for subfamilies admitting invariant algebraic surfaces of degree two. Then for some such subfamilies we construct first integrals of the Darboux type, identifying the systems with one first integral or with two independent first integrals.

In this paper we investigate the simultaneous existence of isochronous centers for a family of quartic polynomial differential systems under four different types of symmetry. Firstly, we find the normal forms for the system under each type of symmetry. Next, the conditions for the existence of isochronous bi-centers are presented. Finally, we study the global phase portraits of the systems possessing isochronous bi-centers. The study shows the existence of seven non topological equivalent global phase portraits, where three of them are exclusive for quartic systems under such conditions.

We investigate the center problem for polynomial maps of the form \( \displaystyle f(x)=-x-\sum_{n=1}^{\infty}a_nx^{n+1} \), arising from algebraic curves \( x+y+\displaystyle \sum_{k=0}^{n}\alpha_{n-k,k}x^{n-k}y^k=0 \). Previous studies of homogeneous polynomial maps of even degree and of mixed terms of degrees 2 and 4 revealed two recurring algebraic families characterizing the existence of a center at the origin: mirror-symmetry conditions and alternating-sum conditions. In this paper, we resolve the next open case involving mixed terms of degrees 2 and 6 and show that the same algebraic structure extends to mixed terms of degrees 2 and arbitrary even number. Using an approach based on the involutive identity \( f(f(x))=x \), we prove that these two families of algebraic conditions completely characterize the existence of a center at the origin. The proof combines algebraic and structural arguments and avoids the use of explicit higher-order focus quantity computations in the general case.

We investigate the center problem for analytic maps implicitly defined by algebraic equations of the form ${\displaystyle F(x,y)=x+y+{\displaystyle \sum _{i+k=2}^n}{\alpha }_{i,k}{x}^i{y}^k=0}$. Previous studies of homogeneous analytic maps of even degree and of mixed terms of degrees 2 and 4 revealed two recurring algebraic families characterizing the existence of a center at the origin: mirror-symmetry conditions and alternating-sum conditions. In this paper, we resolve the next open case involving mixed terms of degrees 2 and 6 and show that the same algebraic structure extends to mixed terms of degrees 2 and arbitrary even number. Using an approach based on the involutive identity $f\left(f\right(x\left)\right)=x$, we prove that these two families of algebraic conditions completely characterize the existence of a center at the origin. The proof combines algebraic and structural arguments and avoids the use of explicit higher-order focus quantity computations in the general case.