= Solution
Let $S(x,y,z)=(-x,-y,z)$. In the first two components of the <vector field>, replacing $x,y$ by their negatives changes the sign of the whole component; in the third, the product $xy$ is unchanged. Thus $F(Sv)=SF(v)$, establishing the <symmetry>, with solutions mapped to solutions by $S$.
At an <equilibrium point>, positivity of $a$ makes $y=x$. The remaining equations give $x(2c-a-z)=0$ and $z=x^2/b$. Hence
$$
\boxed{O=(0,0,0),\qquad P_\pm=(\pm\sqrt{b(2c-a)},\ \pm\sqrt{b(2c-a)},\ 2c-a),}
$$
where the two nonzero <equilibrium points> exist when $0<a<2c$. They merge with the origin at $a_p=2c$, in a <pitchfork bifurcation>.
At the origin the <Jacobian matrix> and its <eigenvalues> are
$$
J_O=\begin{pmatrix}-a&a&0\\c-a&c&0\\0&0&-b\end{pmatrix},\qquad
\lambda_z=-b,\quad\lambda_\pm=\frac{c-a\pm\sqrt{c^2+6ac-3a^2}}2.
$$
At $a=2c$ these are $0,-c,-b$. To see the local <pitchfork bifurcation> dynamics more explicitly put $\delta=2c-a$, $y=x+v$, and use $z=x^2/b+\cdots$ on the <center manifold>. The equations become
$$
\dot x=av,\qquad \dot v=\delta x+(c-a)v-xz,\qquad
\dot z=x^2+xv-bz.
$$
At $\delta=0$, invariance at cubic order gives $v=-x^3/(bc)+\cdots$, hence the reduced equation is
$$
\dot x=2\delta x-\frac2b x^3+\text{higher-order terms}.
$$
It has the supercritical <pitchfork bifurcation> form as $\delta$ increases, with the two branches $x\simeq\pm\sqrt{b\delta}$ <attracting> near onset while the origin becomes <unstable>.
For $0<a<2c$, the <determinant> of the planar block is $a(a-2c)<0$, so the origin has one positive and two negative <eigenvalues> and is a <saddle equilibrium>. At $a=c$, the planar <eigenvalues> are explicitly $\pm c$. For $a>2c$, the planar <trace> $c-a$ is negative and its <determinant> positive, so the origin is <asymptotically stable>. To distinguish its type put
$$
a_*=c\left(1+\frac{2\sqrt3}{3}\right).
$$
For $2c<a<a_*$ both planar <eigenvalues> are real negative, giving a <stable node>. At $a=a_*$ they coincide at $-c/\sqrt3$; the planar block is not a scalar <matrix>, so it is a defective <stable node>. For $a>a_*$ they are <nonreal> with negative <real part>, giving a <stable> spiral in the planar block together with the <stable> third direction. This refines the requested node-versus-saddle classification.
The origin cannot have a <Hopf bifurcation>: a planar imaginary pair would require <trace> zero, namely $a=c$, but the <determinant> there is $-c^2<0$, so the pair is real and of opposite signs.
For either nonzero <equilibrium point> put $s=\pm\sqrt{b(2c-a)}$. The <Jacobian matrix> is
$$
J_\pm=\begin{pmatrix}-a&a&0\\-c&c&-s\\s&s&-b\end{pmatrix}.
$$
Its <characteristic polynomial> is
$$
p(\lambda)=\lambda^3+A\lambda^2+B\lambda+C,\qquad
A=a+b-c,\quad B=bc,\quad C=2ab(2c-a).
$$
A nonzero imaginary <root of an equation> $i\omega$ requires, by separating real and imaginary parts, $\omega^2=B$ and $AB=C$. Equivalently
$$
2a^2-3ac+bc-c^2=0,
$$
so the candidate <Hopf bifurcation> values are
$$
\boxed{a_h=\frac{3c\pm\sqrt{17c^2-8bc}}4.}
$$
At such an admissible value the factorization is $p(\lambda)=(\lambda+A)(\lambda^2+bc)$, giving the pair $\pm i\sqrt{bc}$ and a negative third <eigenvalue> $-A$, since $A=C/B>0$ on the nonzero branches.
The <mathematical model parameter> restrictions matter. A real candidate requires $b/c\le17/8$, and the nonzero <equilibrium points> also require $0<a_h<2c$. The upper-sign value is admissible for $0<b/c<17/8$; the lower-sign value additionally requires $b/c>1$. For fixed $b,c$, <differentiation> of the characteristic equation at the imaginary <root of an equation> gives the <eigenvalue crossing speed>
$$
\left.\frac{d\operatorname{Re}\lambda}{da}\right|_{a_h}
=-\frac{b(4a_h-3c)}{2\{bc+(a_h+b-c)^2\}}.
$$
Thus the <transverse eigenvalue crossing> condition holds except at the coincident-root case $b/c=17/8$, $a_h=3c/4$. At a <transverse eigenvalue crossing>, a generic nonzero <first Lyapunov coefficient> gives a <Hopf bifurcation>. The displayed radical is the spectral threshold; it does not justify an unconditional generic-Hopf claim for all positive <mathematical model parameters>, nor does it by itself exclude a degenerate <Hopf bifurcation> point. For example $b>17c/8$ yields no real candidate at all. The cubic <Routh-Hurwitz stability criterion> gives local <Lyapunov stability> on the side with $A>0$ and $AB-C>0$, where
$$
AB-C=b(2a^2-3ac+bc-c^2).
$$
For the modified equation, the planar block at the origin is $\begin{pmatrix}-a&a\\c&c\end{pmatrix}$, with <determinant> $-2ac<0$ for the stipulated positive <mathematical model parameters>. Thus the origin is always a <saddle equilibrium> for nonzero $a,c$ in that <mathematical model parameter> range. The nonzero <equilibrium points> now satisfy $y=x$, $z=2c$ and $x^2=2bc$, so they are
$$
(\pm\sqrt{2bc},\ \pm\sqrt{2bc},\ 2c).
$$
For $b,c>0$ these branches stay a positive distance from the origin and do not merge with it as $a$ varies. The origin also has no zero <eigenvalue> for $a,c>0$. Therefore \b[the <pitchfork bifurcation> is absent in the modified system].
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