Non-degenerate approximations of a star-shaped Hamiltonian, agreeing with it to first order along two given non-degenerate periodic orbits

Let $S=H^{-1}(1)\subset\mathbb{R}^4$ be a strictly star-shaped level and let $P=(x_P,T_P)$, $Q=(x_Q,T_Q)$ be periodic orbits of $X_H$ on $S$ such that every periodic orbit of $X_H$ on $S$ whose image lies in $x_P(\mathbb{R})\cup x_Q(\mathbb{R})$ (that is, $P$, $Q$ and all their iterates) is non-degenerate. Then there is a sequence of Hamiltonians $H_k:\mathbb{R}^4\to\mathbb{R}$ such that

1. every level $H_k^{-1}(1)$ is strictly star-shaped;
2. every $H_k$ is non-degenerate: every periodic orbit of $X_{H_k}$ on $H_k^{-1}(1)$, prime or multiply covered, is non-degenerate;
3. $H_k=H$ and $dH_k=dH$ at every point of $x_P(\mathbb{R})\cup x_Q(\mathbb{R})$;
4. $H_k\to H$ in $C^\infty_{\mathrm{loc}}$.

$H_k\to H$ in $C^\infty_{\mathrm{loc}}$ means: for every $m\in\mathbb{N}$ and every compact $K\subset\mathbb{R}^4$, $D^mH_k\to D^mH$ uniformly on $K$. A periodic orbit $R=(x,T)$ on $H^{-1}(1)$ (prime or not) is non-degenerate when, for its linearized flow $Y$, the linearized return map on $\xi=TS/\mathbb{R}X_{H}$, written in the global frame $(Z_1,Z_2)$ as the matrix $\varphi(1)$ of linearizedXiPath, does not have $1$ as an eigenvalue. A Hamiltonian is non-degenerate when all its periodic orbits on the level $1$ are.

Only the orbits $P$ and $Q$ are assumed non-degenerate; the other periodic orbits of $X_H$ may be degenerate. In contact language this is the density, relative to a prescribed $1$-jet along two non-degenerate closed orbits, of non-degenerate contact forms $g\lambda$ near $\lambda$ (Kupka–Smale for Reeb flows).