High Resolution (<100pc) Ionization Structures in the ISM of AGN ESO~137-G34

We present a detailed, spatially resolved { Veilleux \& Osterbrock (VO) diagram} analysis of ESO\,137-G034, a Compton-thick Active Galactic Nucleus (AGN), based on narrowband {\it HST} imaging in [O~III], [S~II], H$\alpha$, and H$\beta$ emission lines. The narrowband optical emission exhibits a bi-conical morphology and traces the diffuse X-ray emission observed with \textit{Chandra}. We dissect the Interstellar Medium (ISM) into Seyfert-, LINER-, and H~II-dominated regions with a resolution of $\sim0.1^{\prime\prime}$ ($\sim20$\,pc at $z=0.009$). To { visualize} the fine spatial structure of the { ISM excitation}, we introduce a new parameter: the Seyfert/LINER Index (SLI), defined as the perpendicular distance of each point in the { VO} diagram from the Seyfert/LINER division line. { SLI mapping reveals fine spatial structures in the ISM excitation especially in the Seyfert ionization bi-cones where the statistical significance of our data is higher.} Most of the emission within the bi-cones is Seyfert-like, with SLI values consistent with regions of the { VO} diagram that can be modeled with AGN photoionization. In the North-West Seyfert cone three radial SLI peaks suggest episodic nuclear outbursts, with estimated timescales of a few $10^3$\,yrs and $<100$\,yrs duration. The South-East cone shows inhomogeneous excitation, with a higher SLI region in its inner part, coinciding with a wider region of enhanced soft X-ray emission at the radio lobe inner edge. This region is characterized by line ratios consistent with fast shocks ($>1000$\,km\,s$^{-1}$) induced excitation. The North-West cone is surrounded by a smooth LINER ``cocoon''. The LINER region in the SE cone appears more irregular, with inter-cone LINER points likely shaped by both local ISM structure and shocks. Our findings highlight the complex interplay of different AGN feedback mechanisms in the ISM.