## 1 Introduction  
The World Health Organization (WHO) defines post-COVID-19 syndrome (PCS) as a constellation of symptoms that occur more than 12 weeks after acute infection with SARS-CoV-2, persist for at least 2 months, which cannot be explained by an alternative etiology [1]. The clinical course may be persistent, relapsing, or fluctuating [1]. Symptoms vary widely and can sometimes lead to considerable limitations in daily life and even to incapacity to work [2, 3]. The neurological symptoms associated with PCS are also highly variable and include exemplarily fatigue (37%), brain fog (32%), memory issues (28%), attention disorder (22%), mylalgia (17%), anosmia (12%), dyspnea (10%), and headache (15%) [4]. A heterogeneous and multifactorial pathogenesis of neurological symptoms are assumed [5, 6]. Several possible mechanisms that may occur simultaneously are discussed [5, 6] as follows: (i) direct neurological invasion and injury by the virus [5, 7, 8], (ii) para-infectious autoimmune responses directed against the central nervous system (CNS) [5], (iii) toxic effects of severe systemic COVID-19 disease on the CNS [5, 9] and (iv) persistent endothelial dysfunction and COVID-19-associated coagulopathy [2, 5, 10, 11, 12, 13]. In large autopsy studies, endothelial dysfunction was detected as endothelialitis in the pons, the thalami, juxta-cortically, and in deep white matter, identified by microbleeds, petechial hemorrhages within vessel walls, and perivascular infiltration of T cells and macrophages [5, 14]. The vascular endothelium, as an essential part of the neurovascular unit, is the main regulator of cerebrovascular reactivity (CVR) [15], which is defined as the change in cerebral blood flow in response to a vasoactive stimulus [15, 16, 17]. In a recent study, a significantly lower whole-brain CVR in patients with previous SARS-CoV-2 infection than in never-before-infected participants was described, whereby the CVR was lower in patients with than without neurological PCS [15]. The CVR was measured using arterial spin labeling (ASL) perfusion imaging with acetazolamide stimulus [15]. Several studies using transcranial color Doppler (TCCD) and a breath-holding test also indicated a reduced CVR in patients with acute SARS-CoV-2 infection and post-SARS-CoV-2 conditions [12, 18, 19, 20], although contradictory findings exist [21]. Hypercapnia-triggered functional magnetic resonance imaging (fMRI) is another option to assess CVR, eliminating the need for intravenous acetazolamide stimulation [16, 22, 23, 24]. The increased arterial partial pressure of CO\({}_{2}\) (PaCO\({}_{2}\)) induces cerebral vasodilation and increases cerebral blood flow [6, 16]. The resulting altered ratio of paramagnetic deoxyhemoglobin to diamagnetic oxym hemoglobin in venules leads to changes in magnetic susceptibility and an increase in the blood-oxygen-level-dependent (BOLD) signal [25]. One highly available method to achieve hypercapnia in order to estimate CVR without the need for CO\({}_{2}\) inhalation systems are breath-hold (bh) periods [8, 13, 16, 23, 24, 25]. Bh-fMRI is highly available and has no possible side-effects and risks, such as the injection of acetazolamide, and can, therefore, be used safely in patients and healthy subjects [13]. As an advantage over TCCD, it provides a high spatial resolution and allows comparisons within any region of interest (ROI) after the normalization of the data set [13]. To date, only a few studies investigated CVR in PCS patients with specific neurological symptoms, and only [PERSON] et al. used MRI [6]. The aim of this pilot study was to investigate whether CVR differences can be detected by bh-fMRI in patients with PCS and leading symptoms of fatigue/memory issues/concentration disorder (PCS\({}_{\text{fmc}}\)) or in patients with PCS and leading symptoms of myopathy/neuropathy (PCS\({}_{\text{mn}}\)), and persons previously infected with SARS-CoV-2 without PCS. The second aim was to investigate if there is a correlation between global CVR and white matter changes in patients with PCS.